# Mathematics

4.2 Identify the parameter, Part II. For each of the following situations, state whether the

parameter of interest is a mean or a proportion.

(a) A poll shows that 64% of Americans personally worry a great deal about federal spending and the budget deficit.

(b) 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.

(c) In a survey, high school and college students are asked whether or not they use geolocation

services on their smart phones.

(d) In a survey, internet users are asked whether or not they purchased any Groupon coupons.

(e) In a survey, internet users are asked how many Groupon coupons they purchased over the last year.

4.4 Heights of adults. Researchers studying anthropometry collected body girth measurements

and skeletal diameter measurements, as well as age, weight, height and gender, for 507 physically

active individuals. The histogram below shows the sample distribution of heights in centimeters.

(a) What is the point estimate for the average height of active individuals? What about the

median?

(b) What is the point estimate for the standard deviation of the heights of active individuals?

What about the IQR?

(c) Is a person who is 1m 80cm (180 cm) tall considered unusually tall? And is a person who is

1m 55cm (155cm) considered unusually short? Explain your reasoning.

(d) The researchers take another random sample of physically active individuals. Would you

expect the mean and the standard deviation of this new sample to be the ones given above.

Explain your reasoning.

(e) The samples means obtained are point estimates for the mean height of all active individuals,

if the sample of individuals is equivalent to a simple random sample. What measure do we use

to quantify the variability of such an estimate? Compute this quantity using the data from

the original sample under the condition that the data are a simple random sample.

4.6 Chocolate chip cookies. 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.

(a) 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?

(b) 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 di_erence might be due to chance. What do you think?

4.8 Mental health. Another question on the General Social Survey introduced in Exercise 4.7

is \For how many days during the past 30 days was your mental health, which includes stress,

depression, and problems with emotions, not good?” Based on responses from 1,151 US residents,

the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.

(a) Interpret this interval in context of the data.

(b) What does a 95% confidence level mean in this context?

(c) Suppose the researchers think a 99% confidence level would be more appropriate for this

interval. Will this new interval be smaller or larger than the 95% confidence interval?

(d) If a new survey asking the same questions was to be done with 500 Americans, would the

standard error of the estimate be larger, smaller, or about the same. Assume the standard

deviation has remained constant since 2010.

4.10 Confidence levels. If a higher con_dence level means that we are more confident about

the number we are reporting, why don’t we always report a confidence interval with the highest

possible confidence level?

4.12 Thanksgiving spending, Part I. The 2009 holiday retail season, which kicked off on November 27, 2009 (the day after Thanksgiving), had been marked by somewhat lower self-reported consumer spending than was seen during the comparable period in 2008. To get an estimate of consumer spending, 436 randomly sampled American adults were surveyed. Daily consumer spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% confidence interval based on this sample is ($80.31, $89.11). Determine whether the following statements are true or false, and explain your reasoning.

(a) We are 95% confident that the average spending of these 436 American adults is between

$80.31 and $89.11.

(b) This confidence interval is not valid since the distribution of spending in the sample is right

skewed.

(c) 95% of such random samples would have a sample mean between $80.31 and $89.11.

(d) We are 95% confident that the average spending of all American adults is between $80.31 and

$89.11.

(e) A 90% confidence interval would be narrower than the 95% confidence interval since we don’t

need to be as sure about capturing the parameter.

(f) In order to decrease the margin of error of a 95% confidence interval to a third of what it is

now, we would need to use a sample 3 times larger.

(g) The margin of error for the reported interval is 4.4.

4.14 Age at first marriage, Part I. The National Survey of Family Growth conducted by the

Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy,

infertility, use of contraception, and men’s and women’s health. One of the variables collected on

this survey is the age at first marriage. The histogram below shows the distribution of ages at

first marriage of 5,534 randomly sampled women between 2006 and 2010. The average age at first

marriage among these women is 23.44 with a standard deviation of 4.72

Estimate the average age at _rst marriage of women using a 95% confidence interval, and interpret

this interval in context. Discuss any relevant assumptions.

4.16 Identify hypotheses, Part II. Write the null and alternative hypotheses in words and

using symbols for each of the following situations.

(a) Since 2008, chain restaurants in California have been required to display calorie counts of

each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners

at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant?

(b) Based on the performance of those who took the GRE exam between July 1, 2004 and June 30, 2007, the average Verbal Reasoning score was calculated to be 462. In 2011 the average verbal score was slightly higher. Do these data provide convincing evidence that the average GRE Verbal Reasoning score has changed since 2004?

4.18 Age at first marriage, Part II. Exercise 4.14 presents the results of a 2006 – 2010 survey showing that the average age of women at first marriage is 23.44. Suppose a researcher believes

that this value has increased in 2012, but he would also be interested if he found a decrease. Below

is how he set up his hypotheses. Indicate any errors you see.

4.20 Thanksgiving spending, Part II. Exercise 4.12 provides a 95% confidence interval for the

average spending by American adults during the six-day period after Thanksgiving 2009: ($80.31,

$89.11).

(a) A local news anchor claims that the average spending during this period in 2009 was $100.

What do you think of this claim?

(b) Would the news anchor’s claim be considered reasonable based on a 90% confidence interval?

Why or why not?

4.22 Gifted children, Part I. Researchers investigating characteristics of gifted children col-

lected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children first counted to 10 successfully. Also provided are some sample statistics.

(a) Are conditions for inference satisfied?

(b) Suppose you read on a parenting website that children first count to 10 successfully when they

are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide

convincing evidence that the average age at which gifted children first count to 10 successfully

is different than the general average of 32 months. Use a significance level of 0.10.

(c) Interpret the p-value in context of the hypothesis test and the data.

(d) Calculate a 90% confidence interval for the average age at which gifted children first count to

10 successfully. (e) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.24 Gifted children, Part II. Exercise 4.22 describes a study on gifted children. In this study, along with variables on the children, the researchers also collected data on the mother’s and father’s IQ of the 36 randomly sampled gifted children. The histogram below shows the distribution of mother’s IQ. Also provided are some sample statistics.

(a) Perform a hypothesis test to evaluate if these data provide convincing evidence that the average IQ of mothers of gifted children is different than the average IQ for the population at large, which is 100. Use a significance level of 0.10.

(b) Calculate a 90% confidence interval for the average IQ of mothers of gifted children.

(c) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.26 Find the sample mean. You are given the following hypotheses:

We know that the sample standard deviation is 10 and the sample size is 65. For what sample

mean would the p-value be equal to 0.05? Assume that all conditions necessary for inference are

satisfied.

4.28 Testing for food safety. A food safety inspector is called upon to investigate a restaurant with a few customer reports of poor sanitation practices. The food safety inspector uses a hypothesis testing framework to evaluate whether regulations are not being met. If he decides the restaurant is in gross violation, its license to serve food will be revoked.

(a) Write the hypotheses in words.

(b) What is a Type 1 error in this context?

(c) What is a Type 2 error in this context?

(d) Which error is more problematic for the restaurant owner? Why?

(e) Which error is more problematic for the diners? Why?

(f) As a diner, would you prefer that the food safety inspector requires strong evidence or very

strong evidence of health concerns before revoking a restaurant’s license? Explain your reasoning.

4.30 Car insurance savings, Part I. A car insurance company advertises that customers switching to their insurance save, on average, $432 on their yearly premiums. A market researcher at a competing insurance discounter is interested in showing that this value is an overestimate

so he can provide evidence to government regulators that the company is falsely advertising their

prices. He randomly samples 82 customers who recently switched to this insurance and finds an

average savings of $395, with a standard deviation of $102.

(a) Are conditions for inference satisfied?

(b) Perform a hypothesis test and state your conclusion.

(c) Do you agree with the market researcher that the amount of savings advertised is an overestimate? Explain your reasoning.

(d) Calculate a 90% confidence interval for the average amount of savings of all customers who

switch their insurance.

(e) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.32 Speed reading, Part I. A company offering online speed reading courses claims that students who take their courses show a 5 times (500%) increase in the number of words they can read in a minute without losing comprehension. A random sample of 100 students yielded an average increase of 415% with a standard deviation of 220%. Is there evidence that the company’s claim is false?

(a) Are conditions for inference satisfied?

(b) Perform a hypothesis test evaluating if the company’s claim is reasonable or if the true average improvement is less than 500%. Make sure to interpret your response in context of the hypothesis test and the data. Use α= 0:025.

(c) Calculate a 95% confidence interval for the average increase in the number of words students

can read in a minute without losing comprehension.

(d) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.34 Ages of pennies, The histogram below shows the distribution of ages of pennies at a bank.

The mean age of the pennies is 10.44 years with a standard deviation of 9.2 years. Using the Central Limit Theorem, calculate the means and standard deviations of the distribution of the mean from random samples of size 5, 30, and 100. Comment on whether the sampling distributions shown agree with the values you compute.

4.36 Identify distributions, Part II. Four plots are presented below. The plot at the top is a distribution for a population. The mean is 60 and the standard deviation is 18. Also shown

below is a distribution of (1) a single random sample of 500 values from this population, (2) a

distribution of 500 sample means from random samples of each size 18, and (3) a distribution of

500 sample means from random samples of each size 81. Determine which plot (A, B, or C) is

which and explain your reasoning.

4.38 Stats final scores. Each year about 1500 students take the introductory statistics course at a large university. 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.

(a) Is the distribution of scores on this final exam symmetric, right skewed, or left skewed?

(b) Would you expect most students to have scored above or below 70 points?

(c) Can we calculate the probability that a randomly chosen student scored above 75 using the

normal distribution?

(d) What is the probability that the average score for a random sample of 40 students is above

75?

(e) How would cutting the sample size in half affect the standard error of the mean?

4.40 CFLs. 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.

(a) What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?

(b) Describe the distribution of the mean lifespan of 15 light bulbs.

(c) What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than

10,500 hours?

(d) Sketch the two distributions (population and sampling) on the same scale.

(e) Could you estimate the probabilities from parts (a) and (c) if the lifespans of light bulbs had

a skewed distribution?

4.42 Spray paint. Suppose the area that can be painted using a single can of spray paint is slightly variable and follows a nearly normal distribution with a mean of 25 square feet and a standard deviation of 3 square feet.

(a) What is the probability that the area covered by a can of spray paint is more than 27 square

feet?

(b) Suppose you want to spray paint an area of 540 square feet using 20 cans of spray paint. On

average, how many square feet must each can be able to cover to spray paint all 540 square

feet?

(c) What is the probability that you can cover a 540 square feet area using 20 cans of spray paint?

(d) If the area covered by a can of spray paint had a slightly skewed distribution, could you still

calculate the probabilities in parts (a) and (c) using the normal distribution?