Sampling Distributions

On Tuesday, the bottles of Arizona Iced Tea filled in a plant were supposed to contain an average of $20$ ounces of iced tea. Quality control inspectors sampled $50$ bottles at random from the day’s production. These bottles contained an average of $20$ounces of iced tea.

On a New York–to–Denver flight, $8\%$ of the $125$ passengers were selected for random security screening before boarding. According to the Transportation Security Administration, $10\%$ of passengers at this airport are chosen for random screening. 

Mars, Incorporated, says that the mix of colors in its M&M’S® Milk Chocolate Candies is $24\%$ blue, $20\%$ orange, $16\%$ green,$14\%$ yellow, $13\%$ red, and $13\%$ brown. Assume that the company’s claim is true. We want to examine the proportion of orange M&M’S in repeated random samples of $50$ candies. Which of the graphs that follow could be the approximate sampling distribution of the statistic? Explain your choice

Graph the population distribution. Identify the individuals, the variable, and the parameter of interest. 

Imagine taking an SRS of $50$ $M\&M’S$. Make a graph showing a possible distribution of the sample data. Give the value of the appropriate statistic for this sample. 

The histogram above left shows the intervals (in minutes) between eruptions of the Old Faithful geyser for all $222$ recorded eruptions during a particular month. For this population, the median is $75$ minutes. We used Fathom software to take $500$ SRSs of size $10$ from the population. The $500$ values of the sample median are displayed in the histogram above right. The mean of the $500$ sample median values is 73.5.

 Is the sample median an unbiased estimator of the population median? Justify your answer 

The histogram above left shows the intervals (in minutes) between eruptions of the Old Faithful geyser for all $222$ recorded eruptions during a particular month. For this population, the median is $75$ minutes. We used Fathom software to take $500$ SRSs of size $10$ from the population. The $500$ values of the sample median are displayed in the histogram above right. The mean of the $500$ sample median values is $73.5$.

Suppose we had taken samples of size $20$ instead of size $10$. Would the spread of the sampling distribution be larger, smaller, or about the same? Justify your answer

3. Describe the shape of the

The histogram above left shows the intervals (in minutes) between eruptions of the Old Faithful geyser for all $222$ recorded eruptions during a particular month. For this population, the median is $75$ minutes. We used Fathom software to take $500$ SRSs of size $10$ from the population. The $500$ values of the sample median are displayed in the histogram above right. The mean of the $500$ sample median values is $73.5$.

Sampling distribution. Explain what this means in terms of overestimating or underestimating the population median 

Identify the population, the parameter, the sample, and the statistic in each setting. 

Stop smoking! A random sample of $1000$ people who signed a card saying they intended to quit smoking were contacted nine months later. It turned out that $210$$(21\%)$ of the sampled individuals had not smoked over the past six months.

identify the population, the parameter, the sample, and the statistic in each setting 

Unemployment Each month, the Current Population Survey interviews a random sample of individuals in about $55,000$ U.S. households. One of their goals is to estimate the national unemployment rate. In December $2009, 10.0\%$ of those interviewed were unemployed

For Exercises 1 to 4, identify the population, the parameter, the sample, and the statistic in each setting 

 Hot turkey Tom is cooking a large turkey breast for a holiday meal. He wants to be sure that the turkey is safe to eat, which requires a minimum internal temperature of $165°F$. Tom uses a thermometer to measure the temperature of the turkey meat at four randomly chosen points. The minimum reading in the sample is $170°F$

For Exercises 1 to 4, identify the population, the parameter, the sample, and the statistic in each setting. 

Gas prices How much do gasoline prices vary in a large city? To find out, a reporter records the price per gallon of regular unleaded gasoline at a random sample of $10$ gas stations in the city on the same day. The range (maximum-minimum) of the prices in the sample is $25$ cents

1) State whether it is a parameter or a statistic and (2)use appropriate notation to describe each

 number; for example, p = 0.65. 

 Get your bearings A large container of ball bearings has mean diameter 2.5003 centimeters (cm). This is within the specifications for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have a mean diameter of 2.5009 cm. Because this is outside the                      

For each boldface number in Exercises 5 to 8, (1) state whether it is a parameter or a statistic and (2) use appropriate notation to describe each number; for example, p = 0.65 

Florida voters Florida has played a key role in recent presidential elections. Voter registration records show that 41% of Florida voters are registered as Democrats. To test a random digit dialing device, you use it to call 250 randomly chosen residential telephones in Florida. Of the registered voters contacted, 33% are registered, Democrats 

Unlisted numbers: A telemarketing firm in Los Angeles uses a device that dials residential telephone numbers in that city at random. Of the first $100$ numbers dialed, $48\%$ are unlisted. This is not surprising because $52\%$ of all Los Angeles residential phones are unlisted.

How tall? A random sample of female college students has a mean height of $64.5$ inches, which is greater than the $63$-inch mean height of all adult American women. 

Doing homework A school newspaper article claims that $60\%$ of the students at a large high school did all their assigned homework last week. Some skeptical AP Statistics students want to investigate whether this claim is true, so they choose an SRS of $100$ students from the school to interview. What values of the sample proportion pˆ would be consistent with the claim that the population proportion of students who completed all their homework is $p$  $0.60$? To find out, we used Fathom software to simulate choosing $250$ SRSs of size $n=100$ students from a population in which $p=0.60$. The figure below is a dotplot of the sample proportion $pˆ$ of students who did all their homework.

(a) Is this the sampling distribution of $pˆ$? Justify your answer. 

(b) Describe the distribution. Are there any obvious outliers? 

(c) Suppose that $45$ of the $100$ students in the actual sample say that they did all their homework last week. What would you conclude about the newspaper article’s claim? Explain. 

Tall girls According to the National Center for Health Statistics, the distribution of heights for 16-year-old females is modeled well by a Normal density curve with mean $\mu = 64$ inches and standard deviation $\sigma =2.5$ inches. To see if this distribution applies at their high school, an AP Statistics class takes an SRS of $20$ of the $300$ $16$-year-old females at the school and measures their heights. What values of the sample mean x would be consistent with the population distribution being $N(64, 2.5)$? To find out, we used Fathom software to simulate choosing $250$ SRSs of size $n =20$ students from a population that is $N(64, 2.5)$. The figure below is a dotplot of the sample mean height x of the students in the sample.

(a) Is this the sampling distribution of $x$? Justify your answer. 

(b) Describe the distribution. Are there any obvious outliers? 

(c) Suppose that the average height of the $20$ girls in the class’s actual sample is $x =64.7$. What would you conclude about the population mean height $M$ for the $16$-year-old females at the school? Explain. 

Doing homework Refer to Exercise 9. 

(a) Make a graph of the population distribution given that there are $3000$ students in the school. (Hint: What type of variable is being measured?) 

(b) Sketch a possible graph of the distribution of sample data for the SRS of size $100$ taken by the AP Statistics students. 

Tall girls Refer to Exercise 10. 

(a) Make a graph of the population distribution. 

(b) Sketch a possible graph of the distribution of sample data for the SRS of size $20$ taken by the AP Statistics class. 

Cold cabin? The Fathom screen shot below shows the results of taking $500$ SRSs of $10$ temperature readings from a population distribution that’s $N(50, 3)$ and recording the sample variance $2x$ each time.

(a) Describe the approximate sampling distribution. 

(b) Suppose that the variance from an actual sample is $s^{2}x=25$. What would you conclude about the thermostat manufacturer’s claim? Explain 

Cold cabin? The Fathom screenshot below shows the results of taking $500$ SRSs of $10$ temperature readings from a population distribution that’s $N(50, 3)$ and recording the sample minimum each time.

(a) Describe the approximate sampling distribution. 

(b) Suppose that the minimum of an actual sample is $40°F$. What would you conclude about the thermostat manufacturer’s claim? Explain. 

Run a mile During World War II, able-bodied male undergraduates at the University of Illinois participated in required physical training. Each student ran a timed mile. Their times followed the Normal distribution with mean $7.11$ minutes and standard deviation $0.74$ minute. An SRS of $100$ of these students has mean time $\overline{x}=7.15$ minutes. A second SRS of size$100$ has mean $\overline{x}=6.97$ minutes. After many SRSs, the values of the sample mean x follow the Normal distribution with mean $7.11$ minutes and standard deviation$0.074$ minute.

(a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of x. How is it different from the population distribution? 

Scooping beads A statistics teacher fills a large container with $1000$ white and $3000$ red beads and then mixes the beads thoroughly. She then has her students take repeated SRSs of $50$ beads from the container. After many SRSs, the values of the sample proportion pˆ of red beads are approximated well by a Normal distribution with mean of $0.75$ and standard deviation of $0.06$.

(a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of $pˆ$. How is it different from the population distribution? 

IRS audits The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in each state varies from over 15 million in California to about $240,000$ in Wyoming.

(a) Will the sampling variability of the sample proportion change from state to state if an SRS of $2000$ tax returns is selected in each state? Explain your answer.

(b) Will the sampling variability of the sample proportion change from state to state if an SRS of $1\%$of all tax returns is selected in each state? Explain your answer

Predict the election Just before a presidential election, a national opinion poll increases the size of its weekly random sample from the usual $1500$ people to $4000$ people. 

(a) Does the larger random sample reduce the bias of the poll result? Explain. 

(b) Does it reduce the variability of the result? Explain 

Bias and variability The figure below shows his programs of four sampling distributions of different statistics intended to estimate the same parameter.

(a) Which statistics are unbiased estimators? Justify your answer. (b) Which statistic does the best job of estimating the parameter? Explain. 

A sample of teens A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of $13$- to $16$-year-olds. The researchers will report the mean $x$ from their sample as an estimate of the mean cholesterol level M in this population.

 (a) Explain to someone who knows no statistics what it means to say that $x$ is an unbiased estimator of $\mu$.

(b) The sample result x is an unbiased estimator of the population mean $\mu$ no matter what size SRS the study chooses. Explain to someone who knows no statistics why a large random sample gives more trustworthy results than a small random sample.

A newspaper poll reported that $73\%$ of respondents liked business tycoon Donald Trump. The number $73\%$ is

(a) a population. 

(b) a parameter. 

(c) a sample. 

(d) a statistic. 

(e) an unbiased estimator. 

The name for the pattern of values that a statistic takes when we sample repeatedly from the same population is 

(a) the bias of the statistic. 

(b) the variability of the statistic. 

(c) the population distribution. 

(d) the distribution of sample data. 

(e) the sampling distribution of the statistic. 

If we take a simple random sample of size $n=500$ from a population of size $5,000,000$, the variability of our estimate will be

(a) much less than the variability for a sample of size $n=500$ from a population of size $50,000,000$. 

(b) slightly less than the variability for a sample of size $n=500$ from a population of size $50,000,000$.

(c) about the same as the variability for a sample of size $n=500$ from a population of size $50,000,000$.

(d) slightly greater than the variability for a sample of size $n=500$ from a population of size $50,000,000$.

(e) much greater than the variability for a sample of size $n=500$ from a population of size $50,000,000$. 

Increasing the sample size of an opinion poll will 

(a) reduce the bias of the poll result. 

(b) reduce the variability of the poll result. 

(c) reduce the effect of nonresponse on the poll. 

(d) reduce the variability of opinions. 

(e) all of the above. 

Dem bones (2.2) Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density (BMD). BMD is usually reported in standardized form. The standardization is based on a population of healthy young adults. The World Health Organization (WHO) criterion for osteoporosis is a BMD score that is $2.5$ standard deviations below the mean for young adults. BMD measurements in a population of people similar in age and gender roughly follow a Normal distribution.

(a) What percent of healthy young adults have osteoporosis by the WHO criterion? 

(b) Women aged $70$ to $79$ are, of course, not young adults. The mean BMD in this age group is about 2 on the standard scale for young adults. Suppose that the standard deviation is the same as for young adults. What percent of this older population has osteoporosis? 

Squirrels and their food supply (3.2) Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over $16$ years. Computer output from a least-squares regression on these data and a residual plot.

(a) Give the equation for the least-squares regression line. Define any variables you use. 

(b) Explain what the residual plot tells you about how well the linear model fits the data. 

(c) Interpret the values of $r^2$ and s in context. 

 About $75\%$  of young adult lnternet users $(Ages 18-29)$ watch online viden. Suppose that a sample survey contacts an SRS of $1000$ young adult Internet users and calculates the proportion of $\overbrace{P}$ in this sample who watch online video.

What is the mean of sampling distribution of $\overbrace{P}$ explain

About $75\%$ of young adult internet users $(ages 18-29)$ watch online video. Suppose that a sample survey contacts an SRS of $1000$ young adult Internet users and calculates the proportion of $\overbrace{p}$ in this sample who watch online video.

Find the standard deviation of the sampling distribution of $\overbrace{p}$.check that the $10\%$condition is met.

About $75\%$of young adult internet users $(ages 18-29)$ watch online video. Suppose that a sample survey contacts an SRS of $1000$ young adult Internet users and calculates the proportion of $\overbrace{p}$ in this sample who watch online video.

Is the sampling distribution of $\overbrace{p}$ approximately normal? check that the normal condition is met.

About $75\%$ of young adult Internet users (ages 18 to 29) watch online video. Suppose that a sample survey contacts an SRS of 1000 young adult Internet users and calculates the proportion $\hat{p}$ in this sample who watch online video.

4. If the sample size were 9000 rather than 1000 , how would this change the sampling distribution of $\hat{p}$?

The candy machine Suppose a large candy machine has $15\%$ orange candies. Use Figure 7.13 (page 435) to help answer the following questions.

(a) Would you be surprised if a sample of 25 candies from the machine contained 8 orange candies (that's $32\%$ orange)? How about 5 orange candies ( $20\%$ orange)? Explain.

(b) Which is more surprising: getting a sample of 25 candies in which $32\%$are orange or getting a sample of 50 candies in which $32\%$ are orange? Explain.

The candy machine Suppose a large candy machine has 15% orange candies. Use Figure 7.13 (page 435) to help answer the following questions.

(a) Would you be surprised if a sample of 25 candies from the machine contained 8 orange candies (that's $32\%$orange)? How about 5 orange candies ( orange)? Explain.

(b) Which is more surprising: getting a sample of 25 candies in which $32\%$ are orange or getting a sample of 50 candies in which $32\%$ are orange? Explain.

29. The candy machine Suppose a large candy machine has 45% orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion $\hat{p}$ of orange candies.
(a) What is the mean of the sampling distribution of $\hat{p}$? Why?
(b) Find the standard deviation of the sampling distribution of $\hat{p}$. Check to see if the $10\%$condition is met.
(c) Is the sampling distribution of $\hat{p}$approximately Normal? Check to see if the Normal condition is met.
(d) If the sample size were 50 rather than 25, how would this change the sampling distribution of $\hat{p}$?

The candy machine Suppose a large candy machine has $15\%$ orange candies. Imagine taking an $SRS$ of $25$ candies from the machine and observing the sample proportion $\hat{p}$ of orange candies.

(a) What is the mean of the sampling distribution of $\hat{p}$ ? Why?

(b) Find the standard deviation of the sampling distribution of $\hat{p}$. Check to see if the $10\%$ condition is met.

(c) Is the sampling distribution of $\hat{p}$ approximately Normal? Check to see if the Normal condition is met.

(d) If the sample size were $75$ rather than $25$, how would this change the sampling distribution of $\hat{p}$ 

Airport security The Transportation Security Administration (TSA) is responsible for airport safety. On some flights, TSA officers randomly select passengers for an extra security check before boarding. One such flight had 76 passengers -12 in first class and 64 in coach class. TSA officers selected an SRS of 10 passengers for screening. Let $\hat{p}$ be the proportion of first-class passengers in the sample.

(a) Is the $10\%$ condition met in this case? Justify your answer.

(b) Is the Normal condition met in this case? Justify your answer.

Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses an SRS of 7 tiles. Let $\hat{p}$ be the proportion of vowels in her sample.

(a) Is the $10\%$ condition met in this case? Justify your answer.

(b) Is the Normal condition met in this case? Justify your answer.

Hispanic workers A factory employs 3000 unionized workers, of whom $30\%$are Hispanic. The 15 -member union executive committee contains 3 Hispanics. What would be the probability of 3 or fewer Hispanics if the executive committee were chosen at random from all the workers?

CHALLENGE: See if you can compute the probability using another method.

Studious athletes A university is concerned about the academic standing of its intercollegiate athletes. A study committee chooses an SRS of 50 of the 316 athletes to interview in detail. Suppose that $40\%$ of the athletes have been told by coaches to neglect their studies on at least one occasion. What is the probability that at least 15 in the sample are among this group?

Do you drink the cereal milk? A USA Today Poll asked a random sample of $1012$ U.S. adults what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, $67\%$ said that they drink it. Suppose that $70\%$ of U.S. adults actually drink the cereal milk. Let $\hat{p}$ be the proportion of people in the sample who drink the cereal milk.

(a) What is the mean of the sampling distribution of $\hat{p}$ ? Why?

(b) Find the standard deviation of the sampling distribution of $\hat{p}$. Check to see if the $10\%$ condition is met.

(c) Is the sampling distribution of $\hat{p}$ approximately Normal? Check to see if the Normal condition is met.

(d) Find the probability of obtaining a sample of $1012$ adults in which $67\%$ or fewer say they drink the cereal milk. Do you have any doubts about the result of this poll?

Do you go to church? The Gallup Poll asked a random sample of $1785$ adults whether they attended church or synagogue during the past week. Of the respondents, $44\%$ said they did attend. Suppose that $40\%$ of the adult population actually went to church or synagogue last week. Let $\hat{p}$ be the proportion of people in the sample who attended church or synagogue.

(a) What is the mean of the sampling distribution of $\hat{p}$ ? Why?

(b) Find the standard deviation of the sampling distribution of $\hat{p}$. Check to see if the $10\%$ condition is met.

(c) Is the sampling distribution of $\hat{p}$ approximately Normal? Check to see if the Normal condition is met.

(d) Find the probability of obtaining a sample of $1785$ adults in which $44\%$ or more say they attended church or synagogue last week. Do you have any doubts about the result of this poll?

Do you drink cereal milk? What sample size would be required to reduce the standard deviation of the sampling distribution to one-half the value you found in Exercise 35(b)? Justify your answer.

Do you go to church? What sample size would be required to reduce the standard deviation of the sampling distribution to one-third the value you found in Exercise 36 (b)? Justify your answer.