6.7: Practice (Chapter 6)

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6.1: Sampling Distributions

Identify each of the following symbols
1. $ \mu $
2. $ \hat{p} $
3. $ \sigma $
4. $ \bar{x} $
5. $ p $
6. $ n $
For each of the following situations, state whether the parameter of interest is a mean or a proportion. It may be helpful to examine whether individual responses are numerical or categorical.
1. In a survey, one hundred college students are asked how many hours per week they spend on the Internet.
2. In a survey, one hundred college students are asked: "What percentage of the time you spend on the Internet is part of your course work?"
3. In a survey, one hundred college students are asked whether or not they cited information from Wikipedia in their papers.
4. In a survey, one hundred college students are asked what percentage of their total weekly spending is on alcoholic beverages.
5. In a sample of one hundred recent college graduates, it is found that 85 percent expect to get a job within one year of their graduation date.
For each of the following situations, state whether the parameter of interest is a mean or a proportion.
1. A poll shows that 64% of Americans personally worry a great deal about federal spending and the budget deficit.
2. A survey reports that local TV news has shown a 17% increase in revenue between 2009 and 2011 while newspaper revenues decreased by 6.4% during this time period.
3. In a survey, high school and college students are asked whether or not they use geolocation services on their smart phones.
4. In a survey, internet users are asked whether or not they purchased any Groupon coupons.
5. In a survey, internet users are asked how many Groupon coupons they purchased over the last year.
Explain in your own words what a sampling distribution is.
How many samples of 2 can be made from a population of size 10?
How many samples of 5 can be made from a population of size 6?
What is the difference between the distribution of a population and a sampling distribution of a statistic?
Give an example of a situation where you would want to collect many samples and examine the distribution of their means or proportions.
Why can't we expect every random sample to exactly match the population?
If five different students each collect a sample and calculate a sample mean, will they get the same value? Why or why not?
A nonprofit wants to understand the fraction of households that have elevated levels of lead in their drinking water. They expect at least 5% of homes will have elevated levels of lead, but not more than about 30%. They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions.
1. What is this distribution called?
2. Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning.
3. If the proportions are distributed around 8%, what is the variability of the distribution?
4. What is the formal name of the value you computed in (c)?
5. Suppose the researchers budget is reduced, and they are only able to collect 250 observations per sample, but they can still collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 800 observations?
In a random sample 765 adults in the United States, 322 say they could not cover a $400 unexpected expense without borrowing money or going into debt.
1. What population is under consideration in the data set?
2. What parameter is being estimated?
3. What is the point estimate for the parameter?
4. What is the name of the statistic we use to measure the uncertainty of the point estimate?
5. Compute the value from part (d) for this context.
6. A cable news pundit thinks the value is actually 50%. Should she be surprised by the data?
7. Suppose the true population value was found to be 40%. If we use this proportion to recompute the value in part (e) using p = 04 instead of p, does the resulting value change much?

Suppose you are conducting an experiment where you have 10 trials. In each trial, you flip a coin 4 times.

Complete the following table by determining the proportion of heads for each sample

Sample	Sample Proportion $ \hat{p} $
H H H T
H H T H
H T H T
T T T T
T H T T
H T T T
H T H T
T T H T
H H H H
T H T T

Now complete the following table for the sampling distribution for the sample proportion

Sample Proportion $ \hat{p} $	Frequency	$ P( \hat{p} ) $
0
0.25
0.5
0.75
1

Consider the list of numbers {2,3,4}.

Complete the following table by writing the 9 possible combinations of choosing two numbers, with replacement, from this list and the corresponding sample mean.

Sample	Sample Mean $ \bar{x} $
2, 2
2, 3

Now complete the following table for the sampling distribution for the sample proportion

Sample Mean $ \bar{x} $	Frequency	$ P( \bar{x} ) $
2
2.5
3
3.5
4

The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows: 8.625, 30.25, 27.625, 46.75, 32.875, 18.25, 5, 0.125, 2.9375, 6.875, 28.25, 24.25, 21, 1.5, 30.25, 71, 43.5, 49.25, 2.5625, 31, 16.5, 9.5, 18.5, 18, 9, 10.5, 16.625, 1.25, 18, 12.87, 7, 12.875, 2.875, 60.25, 29.25.
1. Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of ten.
2. In words, describe the distribution of stock prices.
3. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five pieces together until you have ten averages. List those ten averages.
4. Construct a histogram of the distribution of the averages. Start at x = -0.0005. Use bar widths of ten.
5. Does this histogram look like the graph in part c?
6. In one or two complete sentences, explain why the graphs either look the same or look different?

6.2: Mean and Standard Deviation of Sampling Distributions

Identify each of the following symbols
1. $ \mu_{\hat{p}} $
2. $ \mu_{\bar{x}} $
3. $ \sigma_{\hat{p}} $
4. $ \sigma_{\bar{x}} $
Determine if the following statements are True or False:
1. The standard deviation for the distribution of sample means is the same as the standard deviation of the parent distribution.
2. For any distribution with mean $ \mu $ and standard deviation $ \sigma $, if samples of size $ n $ are taken then the mean of the sampling distribution of the means is also $ \mu $.
3. The standard deviation for the distribution of sample means will get smaller as the sample size gets larger.
4. As the sample size increases, the mean of the sampling distribution increases.
5. For sample proportions, the mean of the sampling distribution is equal to the population mean.
We say that a point estimator is unbiased if (choose any):
1. its sampling distribution is centered exactly at the parameter it estimates.
2. the standard deviation of its sampling distribution decreases as the sample size increases.
3. its sampling distribution is normal.
4. its value is always equal to the parameter it estimates.
Which point estimators are unbiased? Which are biased?
How do you determine the mean and standard deviation of the sampling distribution for sample means? Write out the exact formulas and describe the process for using them.
A population has a mean of 120 and a standard deviation of 20. What would be the expected mean and standard error of the sample mean based on samples of size 25?
Why is the standard deviation of the sampling distribution smaller than the standard deviation of the population from which it came?
Explain (with an example) why sample means tend to cluster around the population mean.
What does the horizontal axis of a sampling distribution represent?
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
1. What is the mean of the distribution of sample means?
2. What is the standard deviation of the distribution of sample means?
A nonprofit wants determines the fraction of households that have elevated levels of lead in their drinking water is 23%. They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions.
1. What is the mean of the distribution of sample proportions?
According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample 36 taxpayers.
1. What is the mean of the distribution of sample means?
2. What is the standard deviation of the distribution of sample means?
Let $ x $ be normally distributed with mean $ \mu=26 $ and standard deviation $ \sigma=6 $. Use the given mean and standard deviation to label the empirical rule graphs below.
1. The first graph represents the $ x $ distribution
2. The second graph represents the $ \bar{x} $ distribution. Construct the $ \bar{x} $ distribution using samples of size $ n=49 $.
NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of 17 hours with a standard deviation of 0.8 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 16.7 hours.
1. If the process is working properly, what is the probability of getting a random sample of 30 batteries in which the sample mean lifetime is 16.7 hours or less?
2. Is the company’s claim reasonable?
Your company has a contract to perform preventive maintenance on thousands of air-conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple random sample of 70 units in the city. You plan to budget an average of 1.1 hours per technician to complete the work. Will this be enough time?
John is shopping for wireless routers and is overwhelmed by the number of available options. In order to get a feel for the average price, he takes a random sample of 75 routers and finds that the average price for this sample is $75 and the standard deviation is $25.
1. Based on this information, how much variability should he expect to see in the mean prices of repeated samples, each containing 75 randomly selected wireless routers?
2. A consumer website claims that the average price of routers is $80. Is a true average of $80 consistent with John's sample?
Students are asked to count the number of chocolate chips in 22 cookies for a class activity. They found that the cookies on average had 14.77 chocolate chips with a standard deviation of 4.37 chocolate chips.
1. Based on this information, about how much variability should they expect to see in the mean number of chocolate chips in random samples of 22 chocolate chip cookies?
2. The packaging for these cookies claims that there are at least 20 chocolate chips per cookie. One student thinks this number is unreasonably high since the average they found is much lower. Another student claims the difference might be due to chance. What do you think?

6.3: Central Limit Theorem – Meaning and Implications

State the Central Limit Theorem (CLT) in your own words.
Why is the Central Limit Theorem important in statistics?
What are the requirements on sampling and the population so that the distribution of sample means is approximately normal?
When can we say that the sampling distribution of $ \bar{x} $ is approximately normal?
A population is heavily skewed. What sample size is likely sufficient for the CLT to apply?
Give an example of a situation where the CLT helps you use a normal model for inference.
Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let the average of the 49 races.
1. Does the CLT apply to this situation?
2. Describe the sampling distribution
3. Would you be surprised if a runner completed a marathon in 100 minutes?
The length of songs in a collector’s iTunes album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.
1. Does the CLT apply to this situation?
2. Describe the sampling distribution
3. Would it be likely to find a song that is less than 1 minute long?
The attention span of a two-year-old is exponentially distributed with a mean of about eight minutes. Suppose we randomly survey 60 two-year-olds.
1. Explain why the sampling distribution is not exponential
2. Without calculating, which do you think will be higher and why?
  1. The probability that an individual attention span is less than ten minutes.
  2. The probability that the average attention span for the 60 children is less than ten minutes.
Lifespans of televisions have a mean of 4.7 years and a standard deviation of 1.2 years. Samples of 36 televisions are randomly selected. Will the distribution of sample means be normally distributed? Why or why not?
For students who borrow money to pay for college, the amount of debt for college graduates is normally distributed with mean $37,172 and standard deviation $11,536. Assume that samples of 10 college graduates are randomly selected. Is the distribution of sample means normal? Why or why not?
A random variable is not normally distributed, but it is mound shaped. It has a mean of 18 and a standard deviation of 3.
1. If you take a sample of size 13, can you say what the shape of the sampling distribution for the sample mean is? Why?
2. For a sample of size 13, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary.
3. If you take a sample of size 37, can you say what the shape of the distribution of the sample mean is? Why?
4. For a sample of size 37, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary.
The distribution of weights of US pennies is approximately normal with a mean of 2.5 grams and a standard deviation of 0.03 grams.
1. What is the probability that a randomly chosen penny weighs less than 2.4 grams?
2. Describe the sampling distribution of the mean weight of 10 randomly chosen pennies.
3. What is the probability that the mean weight of 10 pennies is less than 2.4 grams?
4. Sketch the two distributions (population and sampling) on the same scale.
Could you estimate the probabilities from and if the weights of pennies had a skewed distribution?~A manufacturer of compact fluorescent light bulbs advertises that the distribution of the lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000 hours.
1. What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?
2. Describe the distribution of the mean lifespan of 15 light bulbs.
3. What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours?
4. Sketch the two distributions (population and sampling) on the same scale.
5. Could you estimate the probabilities from parts and if the lifespans of light bulbs had a skewed distribution?

6.4: CLT for Means and Proportions

Complete the following table for the difference between sample mean and sample proportion:

Empty table for summarizing the Central Limit Theorem.
	Sample Mean ($ \bar{x} $)	Sample Proportion ($ \hat{p} $)
Population parameter	?	?
Mean of sampling distribution	?	?
Standard error formula	?	?

If a population has proportion $ p = 0.6 $, and you collect a sample of size 100, compute the expected mean and standard error of the distribution of sample proportions.
How do conditions for normal approximation differ between means and proportions?
Explain why sampling distributions are usually narrower than the population distribution.
Determine if the following sampling distributions are normal.
1. n=11 p=0.22
2. n=92 p=0.59
3. n=48 p=0.19
It is known that 64% of Americans personally worry a great deal about federal spending and the budget deficit. Which of the following needs to be satisfied in order to be able to use the Central Limit Theorem to describe the sampling distribution of the proportion of Americans who personally worry a great deal about federal spending and the budget deficit?
1. The sample needs to be random.
2. If sampling without replacement, the sample size needs to be less than 10% of the population size.
3. The sample size should not be less than 50.
4. The expected numbers of those who do and do not personally worry a great deal about federal spending and the budget deficit need to both be at least 10.
Vegetarian college students. Suppose that 8% of college students are vegetarians. Determine if the following statements are true or false, and explain your reasoning.
1. The distribution of the sample proportions of vegetarians in random samples of size 60 is approximately normal since $n \ge 30$.
2. The distribution of the sample proportions of vegetarian college students in random samples of size 50 is right skewed.
3. A random sample of 125 college students where 12% are vegetarians would be considered unusual.
4. A random sample of 250 college students where 12% are vegetarians would be considered unusual.
5. The standard error would be reduced by one-half if we increased the sample size from 125 to 250.
Suppose that 90% of orange tabby cats are male. Determine if the following statements are true or false, and explain your reasoning.
1. The distribution of sample proportions of random samples of size 30 is left skewed.
2. Using a sample size that is 4 times as large will reduce the standard error of the sample proportion by one-half.
3. The distribution of sample proportions of random samples of size 140 is approximately normal.
4. The distribution of sample proportions of random samples of size 280 is approximately normal.
Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let $X$ be the random variable representing the time it takes her to complete one review. Assume $X$ is normally distributed. Let $\bar{X}$ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.
1. What is the mean, standard deviation, and sample size?
2. Find the probability that one review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.
3. Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.
4. What causes the probabilities in parts B and C to be different?
5. Find the 95^th percentile for the mean time to complete one month's reviews. Sketch the graph.
The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.
1. Describe the sampling distribution of sample means
2. How is it possible for the standard deviation to be greater than the average?
3. Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?
The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital.
1. Describe the sampling distribution of sample means
2. Is it likely that an individual stayed more than five days in the hospital? Why or why not?
3. Is it likely that the average stay for the 80 women was more than five days? Why or why not?
4. Which is more likely:
  1. An individual stayed more than five days.
  2. The average stay of 80 women was more than five days.
5. If we were to sum up the women’s stays, is it likely that, collectively they spent more than a year in the hospital? Why or why not?
Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones.
1. Describe the sampling distribution of sample proportions.
2. Is it likely that 85% of adult residents of this city have cell phones?
According to a 2009 Reader's Digest article, people throw away about 9% of what they buy at the grocery store. Assume this is the true proportion and you plan to randomly survey 174 grocery shoppers to investigate their behavior. What is the probability that the sample proportion does not exceed 0.17?
Based on historical data, your manager believes that 27% of the company's orders come from first-time customers. A random sample of 168 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is greater than than 0.21?
Suppose that the efficacy of a certain drug is 0.47. Consider the sampling distribution (sample size n = 93) for the proportion of patients cured by this drug. What is the probability that the sample proportion is between 40% and 50%?
A student polls their school to see if students in the school district are for or against the new legislation regarding school uniforms. They survey 600 students and finds that 480 are against the new legislation.
1. Describe the sampling distribution of sample proportions.
2. In a sample of 600 students, do we expect that more than 200 are for the new legislation?
A housing survey was conducted to determine the price of a typical home in Denver, CO. The mean price of a house was roughly $1.3 million with a standard deviation of $300,000. There were no houses listed below $600,000 but a few houses above $3 million.
1. Is the distribution of housing prices in Denver symmetric, right skewed, or left skewed? Hint: Sketch the distribution.
2. Would you expect most houses in Denver to cost more or less than $1.3 million?
3. Can we estimate the probability that a randomly chosen house in Denver costs more than $1.4 million using the normal distribution?
4. What is the probability that the mean of 60 randomly chosen houses in Denver is more than $1.4 million?
5. How would doubling the sample size affect the standard error of the mean?
Each year about 1500 students take the introductory statistics course at Red Rocks Community College. This year, scores on the final exam are distributed with a median of 74 points, a mean of 70 points, and a standard deviation of 10 points. There are no students who scored above 100 (the maximum score attainable on the final) but a few students scored below 20 points.
1. Is the distribution of scores on this final exam symmetric, right skewed, or left skewed?
2. Would you expect most students to have scored above or below 70 points?
3. Can we calculate the probability that a randomly chosen student scored above 75 using the normal distribution?
4. What is the probability that the average score for a random sample of 40 students is above 75?
5. How would cutting the sample size in half affect the standard error of the mean?

6.5: Approximating the Binomial with the Normal Distribution

Which of the following probabilities can be calculated using the normal approximation to binomial?
1. A Gallup poll suggests that 63% of Americans do not have a great deal of confidence in the mass media to report the news fully, accurately, and fairly. What is the probability that in a random sample of 15 people 9 or more of them have confidence in the mass media?
2. A clothing store offers store credit cards and only about 37% of the credit card holders are regular customers. If we were to randomly sample 10 store credit card holders to conduct a survey, what is the probability at most 4 of the sampled individuals would be regular customers?
3. Roughly 600% of adult Americans drink alcohol. What is the probability that in a random sample of 30 people at least 10 are drinkers?
You are working with a binomial experiment with n=20 trials, and a fixed probability of success of p=15%. Can the binomial distribution associated with this experiment be approximated by a normal distribution?
You are working with a binomial experiment with n=30 trials, and a fixed probability of success of p=40%. Can the binomial distribution associated with this experiment be approximated by a normal distribution?
What two conditions must be satisfied before the binomial distribution can be approximated by the normal distribution?
A coin is flipped 100 times. Use the normal approximation to estimate the probability of getting fewer than 45 heads.
Explain the purpose of the continuity correction.
A basketball player hits 85% of their shots. They're going to take 50 free throws. Use the normal approximation to estimate the probability they make at least 40 shots.
Why might the normal approximation to the binomial be inaccurate when $ n $ is small or $ p $ is near 0 or 1?
In a city, 46 percent of the population favor the incumbent, Dawn Morgan, for mayor. A simple random sample of 500 is taken. Using the continuity correction factor, find the probability that at least 250 favor Dawn Morgan for mayor.
Tony runs a cell phone kiosk in the mall and has 248 potential customers each day. She has a 6% chance to make a sale to each customer. Use continuity correction to normally approximate the following probabilities.
1. The probability that she makes at least 16 sales
2. The probability that she makes less than 14 sales
Suppose in a local Kindergarten through 12^th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through 5. A simple random sample of 300 is surveyed.
1. Find the probability that at least 150 favor a charter school.
2. Find the probability that at most 160 favor a charter school.
3. Find the probability that more than 155 favor a charter school.
4. Find the probability that fewer than 147 favor a charter school.
5. Find the probability that exactly 175 favor a charter school.
Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 96 days. Use the normal approximation to the binomial to calculate the following probabilities. Round the standard deviation to four decimal places.
1. Find the probability that Janice is the driver at most 20 days.
2. Find the probability that Roberta is the driver more than 16 days.
3. Find the probability that Barbara drives exactly 24 of those 96 days.

Practice problems include parts of:

Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

and

OpenIntro Statistics by David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0

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