# Mathematics

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?