Determining confidence levels

Question

Assignment:

1. Find z for each of the following confidence levels. Round to two decimal places.

1. 90% 
2. 95% 
3. 96% 
4. 97% 
5. 98% 
6. 99%

2. For a data set obtained from a random sample, n = 81 and x = 48.25. It is known 

that σ = 4.8.

1. What is the point estimate of μ? Round to two decimal places
2. Make a 95% confidence interval for μ. What is the lower limit? Round to two decimal places.
3. Make a 95% confidence interval for μ. What is the upper limit? Round to two decimal places.
4. What is the margin of error of estimate for part b? Round to two decimal places.

3. Determine the sample size (nfor the estimate of μ for the following.

1. E =2.3, σ = 15.40, confidence level = 99%. Round to the nearest whole number.
2. E =4.1, σ = 23.45, confidence level = 95%. Round to the nearest whole number.
3. E =25.9, σ = 122.25, confidence level= 90%. Round to the nearest whole number.

4. True or False.

a.The null hypothesis is a claim about a population parameter that is assumed to be false until it is declared false.

A. True

B. False

b. An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.

A. True

B. False

c. The critical point(s) divide(s) is some of the area under a distribution curve into rejection and nonrejection regions.

A. True

B. False

d. The significance level, denoted by α, is the probability of making a Type II error, that is, the probability of rejecting the null hypothesis when it is actually true.

A. True

B. False

e. The nonrejection region is the area to the right or left of the critical point where the null hypothesis is not rejected.

A. True

B. False

5. A Type I error is committed when

1. A null hypothesis is not rejected when it is actually false
2. A null hypothesis is rejected when it is actually true
3. An alternative hypothesis is rejected when it is actually true

6. Consider H0: μ = 45 versus H1: μ < 45. A random sample of 25 observations produced a sample mean of 41.8. Using α = .025 and the population is known to be normally distributed with σ = 6.

1. What is the value of z? Round to two decimal places.
2. Would you reject the null hypothesis? 

1. Reject Ho
2. Do not reject Ho

7. The following information is obtained from two independent samples selected from two normally distributed populations.

n1 = 18 x1 = 7.82 σ1 = 2.35

n2 =15  x2 =5.99 σ2 =3.17

A. What is the point estimate of μ1 - μ2? Round to two decimal places.

B. Construct a 99% confidence interval for μ1 - μ2. Find the margin of error for this estimate. 

Round to two decimal places. 

8. The following information is obtained from two independent samples selected from two 

populations.

n1 =650 x1 =1.05  σ1 =5.22

n2 =675 x2 =1.54 σ2 =6.80

Test at a 5% significance level if μ1 is less than μ2.

a) Identify the appropriate distribution to use.

1. t distribution
2. normal distribution

b) What is the conclusion about the hypothesis? 

A. Reject Ho

B. Do not reject Ho

9. Using data from the U.S. Census Bureau and other sources, www.nerdwallet.com estimated that considering only the households with credit card debts, the average credit card debt for U.S. house- holds was $15,523 in 2014 and $15,242 in 2013. Suppose that these estimates were based on random samples of 600 households with credit card debts in 2014 and 700 households with credit card debts in 2013. Suppose that the sample standard deviations for these two samples were $3870 and $3764, respectively. Assume that the standard deviations for the two populations are unknown but equal.

a) Let μ1 and μ2 be the average credit card debts for all such households for the years 2014 and 2013, respectively. What is the point estimate of μ1 - μ2? Round to two decimal places. Do not include the dollar sign.

b) Construct a 98% confidence interval for μ1 - μ2. Round to two decimal places. Do not include the dollar sign. 

1. What is the lower bound? Round to two decimal places.
2. What is the upper bound? Round to two decimal places. 

c) Using a 1% significance level, can you conclude that the average credit card debt for such households was higher in 2014 than in 2013? Use both the p-value and the critical-value approaches to make this test.

A. Reject Ho

B. Do not reject Ho

10. Gamma Corporation is considering the installation of governors on cars driven by its sales staff. These devices would limit the car speeds to a preset level, which is expected to improve fuel economy. The company is planning to test several cars for fuel consumption without governors for 1 week. Then governors would be installed in the same cars, and fuel consumption will be monitored for another week. Gamma Corporation wants to estimate the mean difference in fuel consumption with a margin of error of estimate of 2 mpg with a 90% confidence level. Assume that the differences in fuel consumption are normally distributed and that previous studies suggest that an estimate of sd=3sd=3 mpg is reasonable. How many cars should be tested? (Note that the critical value of tt will depend on nn, so it will be necessary to use trial and error.)