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One of the best ways to learn statistics is to solve practice problems. These problems test your understanding of statistics terminology and your ability to solve common statistics problems. Each problem includes a step-by-step explanation of the solution.

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In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state.

What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

The correct answer is C. For this analysis, let P 1 = the proportion of Republican voters in the first state, P 2 = the proportion of Republican voters in the second state, p 1 = the proportion of Republican voters in the sample from the first state, and p 2 = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n 1 ) = 100, and the number of voters sampled from the second state (n 2 ) = 100.

The solution involves four steps.

• Make sure the sample size is big enough to model differences with a normal population. Because n 1 P 1 = 100 * 0.52 = 52, n 1 (1 - P 1 ) = 100 * 0.48 = 48, n 2 P 2 = 100 * 0.47 = 47, and n 2 (1 - P 2 ) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough.
• Find the mean of the difference in sample proportions: E(p 1 - p 2 ) = P 1 - P 2 = 0.52 - 0.47 = 0.05.

σ d = sqrt{ [ P1( 1 - P 1 ) / n 1 ] + [ P 2 (1 - P 2 ) / n 2 ] }

σ d = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] }

σ d = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

z p 1 - p 2 = (x - μ p 1 - p 2 ) / σ d = (0 - 0.05)/0.0706 = -0.7082

Using Stat Trek's Normal Distribution Calculator , we find that the probability of a z-score being -0.7082 or less is 0.24.

Therefore, the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state is 0.24.

See also: Difference Between Proportions

Step-by-Step Statistics Solutions

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Statistics and Probability Problems with Solutions

Problems on statistics and probability are presented. The solutions to these problems are at the bottom of the page.

• Given the data set 4 , 10 , 7 , 7 , 6 , 9 , 3 , 8 , 9 Find a) the mode, b) the median, c) the mean, d) the sample standard deviation. e) If we replace the data value 6 in the data set above by 24, will the standard deviation increase, decrease or stay the same?
• Find x and y so that the ordered data set has a mean of 42 and a median of 35. 17 , 22 , 26 , 29 , 34 , x , 42 , 67 , 70 , y
• Given the data set 62 , 65 , 68 , 70 , 72 , 74 , 76 , 78 , 80 , 82 , 96 , 101, find a) the median, b) the first quartile, c) the third quartile, c) the interquartile range (IQR).
• The exam grades of 7 students are given below. 70 , 66 , 72 , 96 , 46 , 90 , 50 Find a) the mean b) the sample standard deviation
• Twenty four people had a blood test and the results are shown below. A , B , B , AB , AB , B , O , O , AB , O , B , A AB , A , O , O , AB , B , O , A , AB , O , B , A a) Construct a frequency distribution for the data. b) If a person is selected randomly from the group of twenty four people, what is the probability that his/her blood type is not O?
• When a die is rolled and a coin (with Heads and Tails) is tossed, find the probability of obtaining a) Tails and an even number, b) a number greater 3, c) Heads or an odd number,
• A box contains red and green balls. The number of green balls is 1/3 the number of red balls. If a ball is taken randomly from the box, what is the probability that the ball is red?
• A committee of 6 people is to be formed from a group of 20 people. The committee has to have the number of women double that of the men. In how many ways can this committee be formed if there are 12 men?
• A student's marks in five tests are 36%, 78%, 67%, 88% and 98%. The weights for the five tests are 1, 2, 2, 3, 3 respectively. Find the weighted mean ? of the five tests.
• In a group of 40 people, 10 are healthy and every person the of the remaining 30 has either high blood pressure, a high level of cholesterol or both. If 15 have high blood pressure and 25 have high level of cholesterol, a) how many people have high blood pressure and a high level of cholesterol? If a person is selected randomly from this group, what is the probability that he/she b) has high blood pressure (event A)? c) has high level of cholesterol(event B)? d) has high blood pressure and high level of cholesterol (event A and B)? e) has either high blood pressure or high level of cholesterol (event A or B)? f) Use the above to check the probability formula: P(A or B) = P(A) + P(B) - P(A and B).
• A committee of 5 people is to be formed randomly from a group of 10 women and 6 men. Find the probability that the committee has a) 3 women and 2 men. a) 4 women and 1 men. b) 5 women. c) at least 3 women.
• In a school, 60% of pupils have access to the internet at home. A group of 8 students is chosen at random. Find the probability that a) exactly 5 have access to the internet. b) at least 6 students have access to the internet.
• The grades of a group of 1000 students in an exam are normally distributed with a mean of 70 and a standard deviation of 10. A student from this group is selected randomly. a) Find the probability that his/her grade is greater than 80. b) Find the probability that his/her grade is less than 50. c) Find the probability that his/her grade is between 50 and 80. d) Approximately, how many students have grades greater than 80?

Solutions to the above Problems

• The given data set has 2 modes: 7 and 9
• order data : 3 , 4 , 6 , 7 , 7 , 8 , 9 , 9 , 10 : median = 7
• (mean) : m = (3+4+6+7+7+8+9+9+10) / 9 = 7
• The standard deviation will increase since 24 is further from away from the other data values than 6.
• x = 36 , y = 77
• median = 75
• first quartile = 69
• third quartile = 81
• interquartile range = 81 - 69 = 12
• sample standard deviation ? 18.6 (rounded to 1 decimal place)
• 1 - (7/24) = 17/24 ? 0.71 (rounded to 2 decimal places)
• ? = ? x P(X = x) = 0×0.24 + 1×0.38 + 2×0.20 + 3×0.13 + 4×0.05 = 1.37
• 1) using definition ? = √[ ? (x - ?) 2 P(X = x) ] = √[ (0-1.37) 2 ×0.24 + (1-1.37) 2 ×0.38 + (2-1.37) 2 ×0.2 + (3-1.37) 2 ×0.13 + (4-1.37) 2 ×0.05 ] ? 1.13 (rounded to 2 decimal places) 2) using computing formula ? = √[ ? x 2 P(X = x) - ? 2 ] = √[ 0 2 ×0.24 + 1 2 ×0.38 + 2 2 ×0.2 + 3 2 ×0.13 + 4 2 ×0.05 - 1.37 2 ] ? 1.13 (rounded to 2 decimal places)
• If there 12 men, then there are 20 - 12 = 8 women. The committee has six people with the number of women double that of the men, hence the committee has 4 women and 2 men. The number of ways of selecting 4 women from 8 is given by: 8 C 4 = 70. The number of ways of selecting 2 men from 12 is given by: 12 C 2 = 66. The number of selecting 4 women and 2 men to form the committee is given by : 8 C 4 × 12 C 2 = 70 × 66 = 4620
• ? = ? x i × f i / ? f i ? x i × f i = 1×2 + 2×6 + 3×10 + 4×6 + 5×2 + 6×2 = 90 ? f i = 2 + 6 + 10 + 6 + 2 + 2 = 28 ? = 90 / 28 ? 3.21 (rounded to 2 decimal places)
• Let the marks be: x 1 = 36%, x 2 = 78%, x 3 = 67%, x 4 = 88%, x 5 = 98% and the respective weights be: w 1 = 1, w 2 = 2, w 3 = 2, w 4 = 3, w 5 = 3. The weighted mean = ? x i ×w i / ? w i ? x i ×w i = 36% × 1 + 78%×2 + 67%×2 + 88%×3 + 98%×3 = 884% ? w i = 1 + 2 + 2 + 3 + 3 = 11 weighted mean = 884% / 11 = 80%
• a) Let x be the number of people with both high blood pressure and high level of cholesterol. Hence (15 - x) will be the number of people with high blood pressure ONLY and (25 - x) will be the number of people with high level of cholesterol ONLY. We now express the fact that the total number of people with high blood pressure only, with high level of cholesterol only and with both is equal to 30. (15 - x) + (25 - x) + x = 30 solve for x: x = 10 b) 15 have high blood pressure,hence P(A) = 15/40 = 0.375 c) 25 have high level of cholesterol, hence P(B) = 25/40 = 0.625 d) 10 have both,hence P(A and B) = 10/40 = 0.25 e) 30 have either, hence P(A or B) = 30/40 = 0.75 f) P(A) + P(B) - P(A and B) = 0.375 + 0.625 - 0.25 = 0.75 = P(A or B)
• a) In what follows n C r = n! / [ (n - r)!r! ] and is the number of combinations of n objects taken r at the time and P(A) is the probability that even A happens. There are 16 C 5 ways to select 5 people (committee members) out of a total of 16 people (men and women) There are 10 C 3 ways to select 3 women out of 10. There are 6 C 2 ways to select 2 men out of 6. There are 10 C 3 * 6 C 2 ways to select 3 women out of 10 AND 2 men out of 6. P(3 women AND 2 men) = 10 C 3 * 6 C 2 / 16 C 5 = 0.412087 b) similarly: P(4 women AND 1 men) = 10 C 4 * 6 C 1 / 16 C 5 = 0.288461 c) similarly: P(5 women ) = 10 C 5 * 6 C 0 / 16 C 5 = 0.0576923 (in 6 C 0 the 0 is for no men) d) P(at least 3 women) = P(3 women or 4 women or 5 women) since the events "3 women" , "4 women" and "5 women" are all mutually exclusive, then P(at least 3 women) = P(3 women or 4 women or 5 women) = P(3 women) + P(4 women) + P(5 women) ? 0.412087 + 0.288461 + 0.0576923 ? 0.758240
• a) If a pupil is selected at random and asked if he/she has an internet connection at home, the answer would be yes or no and therefore it is a binomial experiment. The probability of the student answering yes is 60% = 0.6. Let X be the number of students answering yes when 8 students are selected at random and asked the same question. The probability that X = 5 is given by the binomial probability formula as follows: P(X = 5) = 8 C 5 (0.6) 5 (1-0.6) 3 = 0.278691 b) P(X ? 6) = P(X = 6 or X = 7 or X = 8) Since all the events X = 6, X = 7 and X = 8 are mutually exclusive, then P(X ? 6) = P(X = 6) + P(x = 7) + P(X = 8) = 8 C 6 (0.6) 6 (1-0.6) 2 + 8 C 7 (0.6) 7 (1-0.6) 1 + 8 C 8 (0.6) 8 (1-0.6) 0 ? 0.315394
• a) x = 80 , z = (80 - 70)/10 = 1 Probablity for grade to be greater than 80 = 1 - 0.8413 = 0.1587 b) x = 50 , z = (50 - 70)/10 = -2 Probablity for grade to be less than 50 = 0.0228 c) The z-scores for x = 50 and x = 80 have already been calculated above. Probablity for grade to be between 50 and 80 = 0.8413 - 0.0228 = 0.8185 d) 0.1587 * 1000 ? 159 (rounded to the nearest unit)
• a) 500 - (170+90+60+50) = 130 tons of steel/iron was recycled. b) 60/500 = 0.12 = 12% of the total recycled was glass.

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Statistics 110: Probability

Strategic Practice and Homework Problems

Actively solving practice problems is essential for learning probability. Strategic practice problems are organized by concept, to test and reinforce understanding of that concept.  Homework problems  usually do not say which concepts are involved, and often require combining several concepts. Each of the Strategic Practice documents here contains a set of strategic practice problems, solutions to those problems, a homework assignment, and solutions to the homework assignment. Also included here are the exercises from the  book that are marked with an s, and solutions to those exercises. 

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Unit 7: Probability

About this unit, basic theoretical probability.

• Intro to theoretical probability (Opens a modal)
• Probability: the basics (Opens a modal)
• Simple probability: yellow marble (Opens a modal)
• Simple probability: non-blue marble (Opens a modal)
• Intuitive sense of probabilities (Opens a modal)
• The Monty Hall problem (Opens a modal)
• Simple probability Get 5 of 7 questions to level up!
• Comparing probabilities Get 5 of 7 questions to level up!

Probability using sample spaces

• Probability with counting outcomes (Opens a modal)
• Example: All the ways you can flip a coin (Opens a modal)
• Die rolling probability (Opens a modal)
• Subsets of sample spaces (Opens a modal)
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Basic set operations

• Intersection and union of sets (Opens a modal)
• Relative complement or difference between sets (Opens a modal)
• Universal set and absolute complement (Opens a modal)
• Subset, strict subset, and superset (Opens a modal)
• Bringing the set operations together (Opens a modal)
• Basic set notation Get 5 of 7 questions to level up!

Experimental probability

• Experimental probability (Opens a modal)
• Theoretical and experimental probabilities (Opens a modal)
• Making predictions with probability (Opens a modal)
• Simulation and randomness: Random digit tables (Opens a modal)
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Randomness, probability, and simulation

• Experimental versus theoretical probability simulation (Opens a modal)
• Theoretical and experimental probability: Coin flips and die rolls (Opens a modal)
• Random number list to run experiment (Opens a modal)
• Random numbers for experimental probability (Opens a modal)
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Addition rule

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• Addition rule for probability (Opens a modal)
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Multiplication rule for independent events

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The math problem that took nearly a century to solve

UC San Diego mathematicians unlock the secret to Ramsey numbers

University of California - San Diego

Ramsey problems, such as r(4,5) are simple to state, but as shown in this graph, the possible solutions are nearly endless, making them very difficult to solve. 

Credit: Jacques Verstraete / UC San Diego

We’ve all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers  Jacques Verstraete  and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

What was Ramsey’s problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you’re guaranteed to find some kind of order within it — either a set of points with no lines between them  or  a set of points with all possible lines between them (these sets are called “cliques”). This is written as r(s,t) where  s  are the points with lines and  t  are the points without lines.

To those of us who don’t deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called “the theorem on friends and strangers” and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don’t know each other. The answer to r(3,3) is six.

“It’s a fact of nature, an absolute truth,” Verstraete states. “It doesn't matter what the situation is or which six people you pick — you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let’s say you knew the solution to r(5,5) was somewhere between 40-50. If you started with 45 points, there would be more than 10 234  graphs to consider!

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations,” Verstraete explained. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete’s radar for most of his professional career. In fact, he first saw the problem in print in  Erdös on Graphs: His Legacy of Unsolved Problems,  written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered $250 to the first person who could solve it.

“Many people have thought about r(4,t) — it’s been an open problem for over 90 years,” Verstraete said. “But it wasn’t something that was at the forefront of my research. Everybody knows it's hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from  pseudo random graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds — upper and lower limits on the possible answer — tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry. 

“It turned out that the pseudorandom graph we needed could be found in finite geometry,” Verstraete stated. “Sam was the perfect person to come along and help build what we needed.”

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of  t . If you want a party where there will always be four people who all know each other or  t  people who all don’t know each other, you will need roughly t 3  people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t 3  is very close to the exact answer.

The findings are currently under review with the  Annals of Mathematics . 

“It really did take us years to solve,” Verstraete stated. “And there were many times where we were stuck and wondered if we’d be able to solve it at all. But one should never give up, no matter how long it takes.”

Verstraete emphasizes the importance of perseverance — something he reminds his students of often. “If you find that the problem is hard and you're stuck, that means it's a good problem. Fan Chung said a good problem fights back. You can't expect it just to reveal itself.”

Verstraete knows such dogged determination is well-rewarded: “I got a call from Fan saying she owes me $250.”

Annals of Mathematics

10.4007/annals.2024.199.2.8

Method of Research

Experimental study

Article Title

The asymptotics of r(4,t)

Article Publication Date

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Original Source

The independent source for health policy research, polling, and news.

Americans’ Challenges with Health Care Costs

Lunna Lopes , Alex Montero , Marley Presiado , and Liz Hamel Published: Mar 01, 2024

This issue brief was updated on March 1, 2024 to include the latest KFF polling data. 

For many years, KFF polling has found that the high cost of health care is a burden on U.S. families, and that health care costs factor into decisions about insurance coverage and care seeking. These costs and the prospect of unexpected medical bills also rank as the top financial worries for adults and their families, and recent polling shows that lowering out-of-pocket health care costs is by and large the public’s top health care priority. Health care affordability is also one of the top issues that voters want to hear presidential candidates talk about during the 2024 election. This data note summarizes recent KFF polling on the public’s experiences with health care costs. Main takeaways include:

• About half of U.S. adults say it is difficult to afford health care costs, and one in four say they or a family member in their household had problems paying for health care in the past 12 months. Younger adults, those with lower incomes, adults in fair or poor health, and the uninsured are particularly likely to report problems affording health care in the past year.
• The cost of health care can lead some to put off needed care. One in four adults say that in the past 12 months they have skipped or postponed getting health care they needed because of the cost. Notably six in ten uninsured adults (61%) say they went without needed care because of the cost.
• The cost of prescription drugs prevents some people from filling prescriptions. About one in five adults (21%) say they have not filled a prescription because of the cost while a similar share say they have instead opted for over-the-counter alternatives. About one in ten adults say they have cut pills in half or skipped doses of medicine in the last year because of the cost.
• Those who are covered by health insurance are not immune to the burden of health care costs. About half (48%) of insured adults worry about affording their monthly health insurance premium and large shares of adults with employer-sponsored insurance (ESI) and those with Marketplace coverage rate their insurance as “fair” or “poor” when it comes to their monthly premium and to out-of-pocket costs to see a doctor.
• Health care debt is a burden for a large share of Americans. About four in ten adults (41%) report having debt due to medical or dental bills including debts owed to credit cards, collections agencies, family and friends, banks, and other lenders to pay for their health care costs, with disproportionate shares of Black and Hispanic adults, women, parents, those with low incomes, and uninsured adults saying they have health care debt.
• Notable shares of adults still say they are worried about affording medical costs such as unexpected bills, the cost of health care services (including out-of-pocket costs not covered by insurance, such as co-pays and deductibles), prescription drug costs, and long-term care services for themselves or a family member. About three in four adults say they are either “very” or “somewhat worried” about being able to afford unexpected medical bills (74%) or the cost of health care services (73%) for themselves and their families. Additionally, about half of adults would be unable to pay an unexpected medical bill of $500 in full without going into debt.

Difficulty Affording Medical Costs

Many U.S. adults have trouble affording health care costs. While lower income and uninsured adults are the most likely to report this, those with health insurance and those with higher incomes are not immune to the high cost of medical care. About half of U.S. adults say that it is very or somewhat difficult for them to afford their health care costs (47%). Among those under age 65, uninsured adults are much more likely to say affording health care costs is difficult (85%) compared to those with health insurance coverage (47%). Additionally, at least six in ten Black adults (60%) and Hispanic adults (65%) report difficulty affording health care costs compared to about four in ten White adults (39%). Adults in households with annual incomes under $40,000 are more than three times as likely as adults in households with incomes over $90,000 to say it is difficult to afford their health care costs (69% v. 21%). (Source: KFF Health Care Debt Survey: Feb.-Mar. 2022 )

When asked specifically about problems paying for health care in the past year, one in four adults say they or a family member in their household had problems paying for care, including three in ten adults under age 50 and those with lower household incomes (under $40,000). Affording health care is particularly a problem for those who may need it the most as one-third of adults who describe their physical health as “fair” or “poor” say they or a family member had problems paying for health care in the past 12 months. Among uninsured adults, half (49%) say they or a family member in their household had problems paying for health care, including 51% of uninsured adults who say they are in fair or poor health.

The cost of care can also lead some adults to skip or delay seeking services. One-quarter of adults say that in the past 12 months, they have skipped or postponed getting health care they needed because of the cost. The cost of care can also have disproportionate impacts among different groups of people; for instance, women are more likely than men to say they have skipped or postponed getting health care they needed because of the cost (28% vs. 21%). Adults ages 65 and older, most of whom are eligible for health care coverage through Medicare, are much less likely than younger age groups to say they have not gotten health care they needed because of cost.

One in four immigrant adults (22%) say they have skipped or postponed care in the past year, rising to about a third (36%) among those who are uninsured. Seven in ten (69%) of immigrant adults who skipped or postponed care (15% of all immigrant adults) said they did so due to cost or lack of health coverage. (Source: The 2023 KFF/LA Times Survey of Immigrants: Apr.-June 2023 )

Six in ten uninsured adults (61%) say they have skipped or postponed getting health care they needed due to cost. Health insurance, however, does not offer ironclad protection as one in five adults with insurance (21%) still report not getting health care they needed due to cost.

KFF health polling from March 2022 also looked at the specific types of care adults are most likely to report putting off and found that dental services are the most common type of medical care that people report delaying or skipping, with 35% of adults saying they have put it off in the past year due to cost. This is followed by vision services (25%), visits to a doctor’s offices (24%), mental health care (18%), hospital services (14%), and hearing services, including hearing aids (10%). (Source: KFF Health Tracking Poll: March 2022 )

A 2022 KFF report found that people who already have debt due to medical or dental care are disproportionately likely to put off or skip medical care. Half (51%) of adults currently experiencing debt due to medical or dental bills say in the past year, cost has been a probititor to getting the medical test or treatment that was recommended by a doctor. (Source: KFF Health Care Debt Survey: Feb.-Mar. 2022 )

Prescription Drug Costs

For many U.S. adults, prescription drugs are a component of their routine care. More than one in four (28%) adults say it is either “somewhat” or “very difficult” for them to afford to pay for prescription drugs. Affording prescription drugs is particularly difficult for adults who take four or more prescription medications (37%) and those in households with annual incomes under $40,000 (40%). Black and Hispanic adults are also more likely than White adults to say it is difficult for them to afford to pay for prescription drugs. (Source: KFF Health Tracking Poll: July 2023 )

The high cost of prescription drugs also leads some people to cut back on their medications in various ways. About one in five adults (21%) say in the past 12 months they have not filled a prescription because of the cost. A similar share (21%) say they have taken an over-the-counter drug instead of getting a prescription filled – rising to about one third of Hispanic adults (32%) and more than one in four adults (27%) with annual household incomes under $40,000. About one in ten adults say that in the past 12 months they have cut pills in half or skipped doses of medicine due to cost. (Source: KFF Health Tracking Poll: July 2023 )

Health Insurance Cost Ratings

Overall, most insured adults rate their health insurance as “excellent” or “good” when it comes to the amount they have to pay out-of-pocket for their prescriptions (61%), the amount they have to pay out-of-pocket to see a doctor (53%), and the amount they pay monthly for insurance (54%). However, at least three in ten rate their insurance as “fair” or “poor” on each of these metrics, and affordability ratings vary depending on the type of coverage people have.

Adults who have private insurance through employer-sponsored insurance or Marketplace coverage are more likely than those with Medicare or Medicaid to rate their insurance negatively when it comes to their monthly premium, the amount they have to pay out of pocket to see a doctor, and their prescription co-pays. About one in four adults with Medicare give negative ratings to the amount they have to pay each month for insurance and to their out-of-pocket prescription costs, while about one in five give their insurance a negative rating when it comes to their out-of-pocket costs to see a doctor.

Medicaid enrollees are less likely than those with other coverage types to give their insurance negative ratings on these affordability measures (Medicaid does not charge monthly premiums in most states, and copays for covered services, where applied, are required to be nominal.) (Source: KFF Survey of Consumer Experiences with Health Insurance )

Health Care Debt

In June 2022, KFF released an analysis of the KFF Health Care Debt Survey , a companion report to the investigative journalism project on health care debt conducted by KFF Health News and NPR, Diagnosis Debt . This project found that health care debt is a wide-reaching problem in the United States and that 41% of U.S. adults currently have some type of debt due to medical or dental bills from their own or someone else’s care, including about a quarter of adults (24%) who say they have medical or dental bills that are past due or that they are unable to pay, and one in five (21%) who have bills they are paying off over time directly to a provider. One in six (17%) report debt owed to a bank, collection agency, or other lender from loans taken out to pay for medical or dental bills, while similar shares say they have health care debt from bills they put on a credit card and are paying off over time (17%). One in ten report debt owed to a family member or friend from money they borrowed to pay off medical or dental bills.

While four in ten U.S. adults have some type of health care debt, disproportionate shares of lower income adults, the uninsured, Black and Hispanic adults, women, and parents report current debt due to medical or dental bills.

Vulnerabilities and Worries About Health Care and Long-Term Care Costs

A February 2024 KFF Health Tracking Poll shows unexpected medical bills and the cost of health care services are at the top of the list of people’s financial worries, with about three-quarters of the public – and similar shares of insured adults younger than 65 – saying they are at least somewhat worried about affording unexpected medical bills (74%) or the cost of health care services (73%) for themselves and their families. Just over half (55%) of the public say they are “very” or “somewhat worried” about being able to afford their prescription drug costs, while about half (48%) of insured adults say they are worried about affording their monthly health insurance premium.

Worries about health care costs pervade among a majority of adults regardless of their financial situation . Among adults who report difficulty affording their monthly bills, more than eight in ten say they are worried about the cost of health care services (86%) or unexpected medical bills (83%). Among those who report being just able to afford their bills, about eight in ten say they are worried about being able to afford unexpected medical bills (84%) or health care services (83%). And even among adults who say they can afford their bills with money left over, six in ten nonetheless say they are “very” or “somewhat worried” about being able to afford unexpected medical bills (62%) or the cost of health care services (60%) for themselves and their family. (Source: KFF Health Tracking Poll: February 2024 )

Many U.S. adults may be one unexpected medical bill from falling into debt. About half of U.S. adults say they would not be able to pay an unexpected medical bill that came to $500 out of pocket. This includes one in five (19%) who would not be able to pay it at all, 5% who would borrow the money from a bank, payday lender, friends or family to cover the cost, and one in five (21%) who would incur credit card debt in order to pay the bill. Women, those with lower household incomes, Black and Hispanic adults are more likely than their counterparts to say they would be unable to afford this type of bill. (Source: KFF Health Care Debt Survey: Feb.-Mar. 2022 )

Among older adults, the costs of long-term care and support services are also a concern. Almost six in ten (57%) adults 65 and older say they are at least “somewhat anxious” about affording the cost of a nursing home or assisted living facility if they needed it, and half say they feel anxious about being able to afford support services such as paid nurses or aides. These concerns also loom large among those between the ages of 50 and 64, with more than seven in ten saying they feel anxious about affording residential care (73%) and care from paid nurses or aides (72%) if they were to need these services. See The Affordability of Long-Term Care and Support Services: Findings from a KFF Survey for a deeper dive into concerns about the affordability of nursing homes and support services.

• Health Costs
• Racial Equity and Health Policy
• Private Insurance
• Affordability
• High Deductible Plans
• Tracking Poll

Also of Interest

• Health Care Debt In The U.S.: The Broad Consequences Of Medical And Dental Bills
• KFF Health Tracking Poll – March 2022: Economic Concerns and Health Policy, The ACA, and Views of Long-term Care Facilities
• KFF’s Kaiser Health News and NPR Launch Diagnosis: Debt, a Yearlong Reporting Partnership Exploring the Scale, Impact, and Causes of the Health Care Debt Crisis in America
• How Financially Vulnerable are People with Medical Debt?

Automatic Table Maintenance in Microsoft Fabric Warehouse - Checkpointing and Statistics - Part 2

By: Koen Verbeeck   |   Updated: 2024-04-10   |   Comments   |   Related: 1 | 2 | 3 | 4 | 5 | > Microsoft Fabric

We're using the warehouse functionality of Microsoft Fabric, and we've already created a couple of tables. Data is loaded into the tables, and reports are running on top of them. Coming from a SQL Server background, do we have to do the same types of maintenance? For example, in SQL Server, we needed to perform maintenance on tables, such as updating statistics, removing fragmentation, or making sure the log files of the database didn't grow out of bounds. Are these tasks still required?

In Part 1 of this tip , we've established some maintenance is required for the warehouse tables in Microsoft Fabric . However, some parts – like the automatic data compaction of the different Parquet files of the delta table – are automated by the system. In this tip, we'll delve into other aspects of table maintenance:

• Checkpointing of the transaction log, which is also automated.
• Statistics, which are also created and updated automatically but can also be manually created or updated through a script.

In Part 1, we showed how to load some sample data into a warehouse. We're using the same table and transaction log, so it's recommended that you read the first part if you haven't already. You can get a free Fabric trial if you want to follow along. Keep in mind that Microsoft Fabric is a fast-moving product with regular updates, which means some of the functionality may have been expanded, changed, or improved by the time you read this.

Table Maintenance in the Fabric Warehouse

Checkpointing.

Up until this point, we have done the following actions on our sample table (see Part 1 ):

• Creating the table
• Inserting data into the table
• Selecting data from the table (triggering the first data compaction)
• Deleting data
• Selecting data from the table (triggering the second data compaction)

This resulted in five JSON log files for the delta table. You can imagine that if lots of transactions are on the table, many log files are created. Each time we read data from the table, all transaction log files need to be read. When there are many transaction log files (suppose you have a streaming process inserting data regularly), this will become less efficient over time. To solve this problem, the system creates a checkpoint after every 10 transactions. A checkpoint file contains a summary of the previous log files, which means that when data is read, only the checkpoint file and log files created after this checkpoint file need to be read. In other words, instead of reading all transaction log files, only 10 or fewer files need to be read.

It is important to know that there are multiple transaction logs. Two are hidden: the internal transaction logs for the warehouse itself and the SQL analytical endpoint. The only transaction log we can actually see is the delta table transaction log. Checkpointing happens for all three logs, but we can only observe the behavior in the delta log.

Let us look at this process in action. First, let's add some transactions by deleting records. The following statements are run one at a time , resulting in separate transactions for each statement.

At this point, we're at exactly 10 transaction log files. When we run another delete statement, another transaction log file is created.

A checkpoint file wasn't automatically created. One minute later, I ran a SELECT COUNT on the table, and at that time, a checkpoint was created. (This might have been triggered by the SELECT, or there was perhaps an automatic asynchronous process at the same time.):

The contents of _last_checkpoint are the following:

It points to the last known checkpoint and has more information on the size and the number of Parquet files. The last checkpoint file ( 09.checkpoint.parquet) is not a JSON file but a Parquet file. It's a bit hard to read. (I opened the file with a Parquet file reader, and it showed mostly empty data because there are a lot of nested structures in it). But, if you open it with Power Query in Power BI Desktop, you can get some sense of the data that's in there:

It's a condensed version of all the previous log files so that by reading only this single checkpoint file, the system can construct a correct transaction log history of the table.

Just like in SQL Server, the query engine of the Fabric warehouse uses statistics to produce an optimal plan (the one with the least amount of estimated work). To have efficient query performance, it's important statistics are up-to-date. Again, similar to the SQL Server database engine, you can create statistics manually or have them created and updated automatically. Since the concepts are similar, you can learn more about SQL Server statistics in the following tips:

• Importance of Update Statistics in SQL Server
• SQL Server Auto Update and Auto Create Statistics Options

Using our sample table, let's dig a bit deeper into its statistics. With the following query (retrieved and adapted from the documentation ), we can find out which automatically created statistics exist for our table:

The following results are returned:

The first statistic with the cardinality estimation was created when the table was populated with data. (You can cross-check with the transaction log files timestamps but add one hour since this query returns the dates in UTC). The statistics for the columns doLocationId , improvementSurcharge , puLocationId, and storeAndFwdFlag were created after the first automatic data compaction. The statistic for IpepPickupDatetime was created before the first big delete statement. This last one is a histogram statistic, just like in SQL Server .

With DBCC SHOW_STATISTICS, we can get more information about this histogram:

This statistic is a bit out-of-date since we deleted about 1/5th of the table. With the following statement, we can update it (if you don't specify a specific statistic, everything should be updated):

When we run the DBCC statement again, we can see the histogram is now up to date:

• If you haven't already, check out Part 1 of this tip , where the sample data is loaded into the warehouse and where automatic data compaction is discussed.
• If you want to learn more about the performance impact statistics can have, check out the tip: How Incorrect SQL Server Table Statistic Estimates Can Cause Slow Query Execution .
• SQL Server itself also has a concept of checkpoints to write pages from the buffer cache to disk. You can learn more about this in the tip: Indirect Checkpoints in SQL Server 2012 .
• You can find more Fabric tips in this overview .

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What are Warehouses in Microsoft Fabric?

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E | Solution Sheets

Hypothesis testing with one sample.

Class Time: __________________________ Name: _____________________________________

• H 0 : _______
• H a : _______
• In words, CLEARLY state what your random variable X ¯ X ¯ or P ′ P ′ represents.
• State the distribution to use for the test.
• What is the test statistic?
• What is the p -value? In one or two complete sentences, explain what the p -value means for this problem.
• Alpha: _______
• Decision: _______
• Reason for decision: _______
• Conclusion: _______

Hypothesis Testing with Two Samples

• In words, clearly state what your random variable X ¯ 1 − X ¯ 2 X ¯ 1 − X ¯ 2 , P ′ 1 − P ′ 2 P ′ 1 − P ′ 2 or X ¯ d X ¯ d represents.
• What is the p -value? In one to two complete sentences, explain what the p-value means for this problem.
• In complete sentences, explain how you determined which distribution to use.

The Chi-Square Distribution

Class Time: __________________________ Name: ____________________________________

• What are the degrees of freedom?
• What is the p -value? In one to two complete sentences, explain what the p -value means for this problem.

F Distribution and One-Way ANOVA

• df ( n ) = ______ df ( d ) = _______
• What is the p -value?

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