26 Jan a. Find the 99?% confidence

Week 3 Homework

Question 1

Let the random variable X follow a normal distribution with muequals60 and sigma2equals49.

a.

Find the probability that X is greater than 70.

b.

Find the probability that X is greater than 45 and less than 75.

c.

Find the probability that X is less than 65.

d.

The probability is 0.3 that X is greater than what? number?

e.

The probability is 0.03 that X is in the symmetric interval about the mean between which two? numbers?

Question 2

Given a random sample of size of nequals900 from a binomial probability distribution with Pequals0.50?, complete parts? (a) through? (e) below.

a. Find the probability that the number of successes is greater than 500.

b. Find the probability that the number of successes is fewer than 430.

c. Find the probability that the number of successes is between 440 and 465.

d. With probability 0.10?, the number of successes is fewer than how? many?

e. With probability 0.06?, the number of successes is greater than how? many?

Question 3

The fuel? consumption, in miles per? gallon, of all cars of a particular model has a mean of 20 and a standard deviation of 5. The population distribution can be assumed to be normal. A random sample of these cars is taken. Complete parts a and b below.

a. Find the probability that the sample mean fuel consumption will be fewer than 19 miles per gallon if samples of 1?, 4?, and 9 are taken.

The probability for a sample of 1 observation is

The probability for a sample of 4 observations is

The probability for a sample of 9 observations is

b. Explain why the three answers in part? (a) differ in the way they do. Draw a graph to illustrate your reasoning.

As the sample size? increases, the mean of the sampling distribution of Upper X overbar —————

and the standard deviation ————–?Therefore, the sample means tend to —————————-the population mean. Since 19 is less than the population? mean, the portion of the sampling distribution of Upper X overbar that represents sample means fewer than 19 ——————- as the sample size increases.

Which graph shows the correct relationship between the sampling distributions of Upper X overbar for two samples of n 1 and n 2 ?observations, where n 2 is larger than n 1 and x overbar is a sample mean less than the population? mean?

Question 4

Suppose that 65?% of all tax returns lead to a refund. A random sample of 100 tax returns is taken.

a. What is the mean of the distribution of the sample proportion of returns leading to? refunds?

b. What is the variance of the sample? proportion?

c. What is the standard error of the sample? proportion?

d. What is the probability that the sample proportion exceeds 0.75??

Question 5

A supplier to automobile manufacturers wants to be sure that the leak rate? (in cubic centimeters per? second) of transmission oil coolers? (TOCs) meets the established specification limits. A random sample of 10 TOCs is? tested, and the leak rates are shown below.

0.043??0.040??0.053??0.043??0.052??0.056??0.043??0.058??0.048??0.053

a. Is there evidence that the data are not normally? distributed?

b. Find a minimum variance unbiased point estimate of the population mean.

c. Use an unbiased estimation procedure to find a point estimate of the variance of the sample mean

Question 6

A clinic offers a? weight-loss program. The table below gives the amounts of weight? loss, in? pounds, for a random sample of 20 of its clients at the conclusion of the program. Assume that the data are normally distributed. Complete parts? (a) and? (b).

23

13

29

18

7

8

17

15

16

20

14

6

12

10

18

22

11

17

15

22

a. Find a 90?% confidence interval for the population mean.

Without doing the? calculations, explain whether a 98?% confidence interval for the population mean would be wider? than, narrower? than, or the same as that found in part? (a). Choose the correct answer below.

A.

It will be wider because the reliability factor will be larger for a 98?% confidence interval than for a 90?% confidence interval.

B.

It will be wider because the reliability factor will be smaller for a 98?% confidence interval than for a 90?% confidence interval.

C.

It will be the same because the confidence interval is being calculated for the same data set.

D.

It will be narrower because the reliability factor will be smaller for a 98?% confidence interval than for a 90?% confidence interval.

Question 7

Assume simple random sampling. Calculate the? 95% confidence interval estimate for the population mean for each of the following.

N=1600?;

n=70?;

s=10?;

x =140

b. nbspb.

N=1625?;

n=80?;

s2equals=64

x =232 2

c. nbspc.

N=5200?;

n=250?;

s2equals=121

x=57.4

Question 8

How large of a sample is needed to estimate the mean of a normally distributed population of each of the? following?

a. MEequals2?; sigmaequals43?; alphaequals0.01

b. MEequals4?; sigmaequals43?; alphaequals0.01

c. Compare and comment on your answers to parts? (a) and? (b).

a. n=

b. n=

c. Choose the correct answer below.

A.

Since the standard deviation and significance level are the same in? (a) and? (b), the required sample sizes are equal in? (a) and? (b).

B.

Holding all else? constant, increasing the sampling error increases the required sample size.

C.

Holding all else? constant, increasing the sampling error decreases the required sample size.

Question 9

A dependent random sample from two normally distributed populations gives the results shown below. Complete parts a and b below.

n=1111????????????

d=28.328.3????????????

sd=1.91.9

a. Find the 99?% confidence interval for the difference between the means of the two populations.

b. Find the margin of error for a 99?% confidence interval for the difference between the means of the two populations.

Question 10

From a random sample of 7 students in an introductory finance class that uses? group-learning techniques, the mean examination score was found to be 76.64 and the sample standard deviation was 2.8. For an independent random sample of 9 students in another introductory finance class that does not use? group-learning techniques, the sample mean and standard deviation of exam scores were 70.46 and 8.5?, respectively. Estimate with 90?% confidence the difference between the two population mean? scores; do not assume equal population variances

The 90?% confidence interval is from a lower limit of ——-to an upper limit of ———–