Mat 240 Homework Complete Solution

Question: #1573

All answers should be rounded to three decimal places unless otherwise indicated.

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1. Twelve percent of all credit card holders eventually become delinquent. Find the probability in a random sample of 20 credit card holders that

a) at least 8 are delinquent.

b) exactly 6 are delinquent.

2. The following table lists the probability distribution of the number of

Televisions owned by families in a city.

x	4	5	6	7	8
P(x)	.42	.05	.16	.22	.15

a) Verify that this is a probability distribution.

b) Find the mean number of televisions owned by a family in this city.

c) Find the standard deviation of the number of televisions owned by a family in this city.

d) Find the probability that a family in the city owned less than six TV’s.

3. Eighty percent of customers who use a smart phone have purchased a second smart phone.

a) Find the probability that in a random sample of 18 customers who use a smart phone, that at most 7 have purchased a second smart phone.

b) In a random sample of 18 customers who use a smart phone, would it be unusual to find that at most 7 have purchased a second smart phone? Why?

4. The mean number of minutes that a person from the Travel Math Team spends riding in their car during a week is 382 minutes. The standard deviation is 54 minutes. If a person from the Travel Math Team is randomly selected, find the probability that he or she will spend the following amount of time in their car. Assume a normal distribution.

a) Less than 300 minutes

b) More than 405 minutes

c) There is an award given to people from the Travel Math Team that spent the most time in their cars during a week. If this award is given to the top 10% of amount of time spent in their car, find the number of minutes for this cut off.

5. When surveying for brand recognition, 18% of consumers recognized Pepsi.

A new survey of 1500 randomly selected consumers is to be conducted.

Find the expected number who recognizes the Pepsi brand name.
Would it be unusual to find 241 consumers who recognize the Pepsi brand name

out of 1500 consumers? Why?

6. The mean incubation time of fertilized goose eggs kept at 100.5 degrees in a still-air incubator is 21 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 2.5 days.

a) Determine the incubation times that make up the middle 95% of fertilized goose eggs given that the mean is 21 days and the standard deviation is 2.5 days. (Note: this is not a sample, it is the population in general.)

b) If a sample of 8 fertilized goose eggs is taken, the sample size is very small. How do we know that the shape of the distribution of is approximately normally distributed?

c) Find the probability that in a sample of 8 fertilized goose eggs the mean incubation time is 19 days or less.

d) What might you conclude if a random sample of 8 fertilized goose eggs had a mean incubation time of 19 days or less?

7. The National Center of Education Statistics surveyed 1875 college graduates about the lengths of time required to earn their bachelor’s degrees. The mean is 5.25 years.

Assume σ = 1.78 years. Based on this data, construct the 90% confidence interval for the mean time required by all college graduates.

8. A Gallup poll conducted December 11-14, 2006 found that 604 of 1,010 Americans own at least one pet.

Verify that the requirements for the constructions of a confidence interval for p are satisfied.
Construct a 95% confidence interval for the proportion of Americans who own at least one pet.
Construct a 98% confidence interval for the proportion of Americans who own at least one pet.
What is the effect of increasing the level of confidence on the width of the interval?
What sample size is required if you want to estimate the population proportion given a confidence level of 90%? (You are willing to accept an error of 3%.)

9. A random sample of 42 women results in a mean height of 64.85 in with a standard deviation of 3.6 inches. Construct the 98% confidence interval for the mean height of women.

10. Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let µ denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the sample of 50 results in a 95% confidence interval for µ of (7.8, 9.4).

Would a sample size of 75, at the same 95% confidence level, provide a narrower or wider interval? Explain your answer.

Consider the following statement:

There is a 95% chance that µ is between 7.8 and 9.4.

Is this statement correct? Why or why not?

11. The “Team” at work was given the opportunity to improve the completion rate, which at this time is p = .78. Three smaller teams took the challenge to improve the completion rate and offered their ideas up to the work force. Samples were taken from each team’s efforts and the following information was recorded:

H₀: p = .78 H₁: p > .78

Team A: P-value = .0678 Team B: P-value = .8432 Team C: P-value = .0229

If based on α = .05 level of significance, were any of the above teams effective at improving the completion rate? Explain.

12. An engineered has studied the safety of bridges in the US. In a study of 312 bridges, the engineer found that only 78 were safe for daily use.

a) Based on these results construct a 95% confidence interval for the proportion of all such bridges.

b) Before this study, an expert believed that there was a .45 probability of bridges in the US being safe for daily use. Based on the study above, do you agree? Why or why not?

13. A growing concern at of employers is time spent in activities like surfing the Internet and emailing friends during work hours. A CEO of a large company wants to determine whether the average amount of wasted time during an eight-hour work day is less than the reported time of 120 minutes that he read in the paper. Each person in a random sample of 10 employees was contacted and asked about daily wasted time at work – they were guaranteed anonymity. The resulting data are as follows:

108 112 117 130 111 131 113 113 105 128

Do these data provide evidence that the mean wasted time for this company is less than 120minutes? The normal plot and box plot show normal behavior and on outliers. Let the significance level of 0.05.

a) H₀: b) Check that conditions for normality are met

H₁:

c) Test used, if any: d) p-Value

e) Decide whether to reject or fail to reject the null hypothesis.

f) Interpret the decision in the context of the original claim.

14. According to several studies, the average amount of sugar in a serving size of breakfast cereal is 15g with a σ = 2.6g. Peter has created a new line of breakfast cereal and claims that his cereals have way less sugar in them. He takes a random sample of 35 of his breakfast cereals and finds the mean amount of sugar in a serving size of his breakfast cereals is 13.9g. At a significance level of .01, is there significant evidence to support Peter’s claim?

a) H₀: b) Check that conditions for normality are met.

H₁: If conditions are not met, then check here.

c) Test used, if any: d) p-Value

e) Decide whether to reject or fail to reject the null hypothesis.

f) Interpret the decision in the context of the original claim.

15. According to a previous survey, 58% of females aged 15 years of age and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 years of age and older and find that 285 are living alone. At a significance level of 0.05, is there enough evidence to support the sociologist’s claim?

a) H₀: b) Check that conditions for normality are met

H₁:

If conditions are not met, then check here.

c) Test used, if any: d) p-Value

e) Decide whether to reject or fail to reject the null hypothesis.

f) Interpret the decision in the context of the original claim.

16. A study of husbands and wives was conducted. The question of interest was the temperature they enjoyed their home to be. The husbands were asked to give the temperature he would keep his house that day and then the wives were asked the same question. The results below show a paired difference test. At a significance level of .01, do husbands like a colder house? Why or why not?

Paired T for Husband - Wife

N Mean StDev SE Mean

Before 55 208.71 1.11 0.42

After 55 206.57 3.05 1.15

Difference 57 2.14 3.18 1.20

99% lower bound for mean difference: -0.20

T-Test of mean difference = 0 (vs < 0): T-Value = -1.78 P-Value = 0.031

a) H₀: b) Check that conditions for normality are met

H₁:

If conditions are not met, then check here.

c) Test used, if any: (If you had to do this on your graphing calculator, which test would you use.) Would the data be dependent or indedpendent?

d) p-Value

e) Decide whether to reject or fail to reject the null hypothesis.

f) Interpret the decision in the context of the original claim.

17. According to the Crown ATM Network, the mean ATM withdrawal is $67, with a standard deviation of $11. PayEase, Inc., manufacturers an ATM that allows one to pay bills (electric, water, parking tickets, and so on), as well as withdraw money. A review of 19 withdrawals shows the mean withdrawal is $73 from a PayEase ATM machine. Do people withdraw more money from a PayEase ATM machine?

a) H₀: b) Check that conditions for normality are met

H₁:

If conditions are not met, then check here.

c) Test used, if any: d) p-Value

e) Decide whether to reject or fail to reject the null hypothesis.

f) Interpret the decision in the context of the original claim.

18. The following Minitab printout shows a simple linear regression equation where the time taken to study (in minutes per week) can be used to predict the grade in a Statistic’s class . Refer to the table below. Is the model significant at a 5% level of significance. Why or why not?

The regression equation is

Grade = 52.8 + .15 Time

Predictor Coef SE Coeff T P

Constant 52.8 35.19 3.05 .013

Time 3.08 0.031 4.91 .026

19. From the model above, use the regression equation and estimate the grade in a Statistic’s class for a student who studies 212 minutes per week.

20. Prosthodentists specialize in the restoration of oral function, including the use of dental implants, veneers, dentures, and crowns. Since repairing chipped veneer is less costly and time consuming than complete restoration, a researcher wanted to compare the shear bond strength of different repair kits for repairs of chipped porcelain veneer in fixed prosthodontics. He randomly divided 20 porcelain specimens into four treatment groups. Group 1 specimens used the Cojet system, group 2 used the Silistor system, group 3 used the Cimara system and group 4 specimens used the Ceramic Repair system. At the conclusion of the study, shear bond strength (in megapascals, MPa) was measured according to ISO 10477. The data was found to verity the requirements to perform a one-way ANOVA test. The results are given below.

Since at least 1 mean is different based on the output below, the p-value is less than .05, decide which test kit(s) has a significantly different bonding strength. Test at the .05 level of significance.

Solution: #1559

Mat 240 Homework Complete Solution

Here, X = Number of delinquent out of 20 credit card holders~Binomial(n=20,p=0.12) a) at least 8 are delinquent. P(X≥8...