# Statistics | Probability homework help

## Practice Set 4

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.

3. t      distribution
4. 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.)

1.  After  you have reviewed Section 8.2 on Experiment, Outcome and Sample                         Space, respond to any one of the application problems –  8.19 ~ 8.27.
2.  After you have reviewed Section 8.2 on Experiment, Outcome and Sample Space, discuss in your own words (with examples):
a. What is the point estimator of the population mean, μ? How would you calculate the margin of error for an estimate of μ?b. Explain the various alternatives for decreasing the width of a confidence interval. Which is the best alternative?c. Briefly explain how the width of a confidence interval decreases with an increase in the sample size. Give an example.d. Briefly explain how the width of a confidence interval decreases with a decrease in the confidence level. Give an example.
3.  After you have reviewed Section 8.1 on Estimation, Point Estimate, and  Interval Estimate discuss briefly in your own words (with examples) the  meaning of an estimator and an estimate.Explain with your own examples, what is a point estimate and an interval estimate.
4. After you have reviewed the Section 9.1 on Hypothesis Testing respond to  the following application problems: 9.7 and 9.8.
5.  After you have reviewed Section 9.2 on Hypothesis Testing when the standard deviation is known, explain in your own words,  a. What are the five steps of a test of hypothesis using the critical value approach? Explain briefly.b. What does the level of significance represent in a test of hypothesis? Explain. Respond to one of the exercises 9.14 ~ 9.24.
6.   After you have reviewed Section 9.2 on Hypothesis Testing when the  standard deviation is known, respond to one of the application problems:  9.25 ~  9.33.
7.   After you have reviewed Section 9.1 on Hypothesis Testing, briefly  explain the meaning of each of the following terms (using your own  words) with examples of your own:

1. Null hypothesis
2. Alternative hypothesis
3. Critical point(s)
4. Significance level
5. Non rejection region
6. Rejection region
7. Tails of a test
8. Two types of errors

8.  State the Null and Alternate Hypotheses for each of the following situations.Describe the two types of errors that can be made in each case. Introductory Statistics, Saylor Academy Publishing, 2012.
Hypothessis Examples.docx (532k)
11.  Read  about the two types of errors we sometime make in Section 9.1.3 Two                 Types of Errors (page 348 onwards). Sometimes punishing an  innocent person,                and sometime letting go a criminal.                 Also watch this  instructional video by clicking on the Lynda.com® link.                  Also, watch the following videos:  https://www.youtube.com/watch?v=FHT6e_mdGoU   https://www.youtube.com/watch?v=VFMcGdWp0MQ                 Share and discuss some of your own examples of the two  types of errors. Why                is it important to learn about the  two types of errors?Attachments areaPreview YouTube video Type I and Type II Errors  Preview YouTube video Statistics 101: Type I and Type II Errors  Attachments areaPreview YouTube video Type I and Type II Errors  Preview YouTube video Statistics 101: Type I and Type II Errors  