# STA 250-Let x be a random variable representing

Question 5Let x be a random variable representing dividend yield of bank stocks. We may assumethat x has a normal distribution with ? = 2.3%. A random sample of 10 bank stocks gave thefollowing yields (in percents).5.7 4.8 6.0 4.9 4.0 3.4 6.5 7.1 5.3 6.1The sample mean is x = 5.38%. Suppose that for the entire stock market, the mean dividendyield is ? = 4.5%. Do these data indicate that the dividend yield of all bank stocks is higherthan 4.5%? Use ? = 0.01.(a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or twotailed test? Which one from below.H0: ? = 4.5%; H1: ? ? 4.5%; two-tailedH0: ? = 4.5%; H1: ? > 4.5%; right-tailedH0: ? > 4.5%; H1: ? = 4.5%; right-tailedH0: ? = 4.5%; H1: ? < 4.5%; left-tailed(b) What sampling distribution will you use? Explain the rationale for your choice ofsampling distribution.Which one from below …vThe Student’s t, since we assume that x has a normal distribution with known ?.The standard normal, since we assume that x has a normal distribution with unknown ?.The Student’s t, since n is large with unknown ?.The standard normal, since we assume that x has a normal distribution with known ?.What is the value of the sample test statistic? (Round your answer to two decimalplaces.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) 6. The price to earnings ratio (P/E) is an important tool in financial work. A random sample of14 large U.S. banks (J. P. Morgan, Bank of America, and others) gave the following P/Eratios†.24 16 22 14 12 13 17 22 15 19 23 13 1118The sample mean is x? 17.1. Generally speaking, a low P/E ratio indicates a “value” or bargain stock. Suppose arecent copy of a magazine indicated that the P/E ratio of a certain stock index is ? = 18.Let x be a random variable representing the P/E ratio of all large U.S. bank stocks. Weassume that x has a normal distribution and ? = 5.3. Do these data indicate that the P/Eratio of all U.S. bank stocks is less than 18? Use ? = 0.01.(a) What is the level of significance?State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, ortwo-tailed test?H0: ? = 18; H1: ? > 18; right-tailedH0: ? = 18; H1: ? < 18; left-tailedH0: ? ? 18; H1: ? = 18; two-tailedH0: ? = 18; H1: ? ? 18; two-tailed(b) What sampling distribution will you use? Explain the rationale for your choiceof sampling distribution.The Student’s t, since n is large with unknown ?.The Student’s t, since we assume that x has a normal distribution with known ?.The standard normal, since we assume that x has a normal distribution with unknown ?.The standard normal, since we assume that x has a normal distribution with known ?.What is the value of the sample test statistic? (Round your answer to two decimalplaces.)(c) Find (or estimate) the P-value. (Round your answer to four decimal places.) 9.A random sample of 51 adult coyotes in a region of northern Minnesota showed the averageage to be x = 2.01 years, with sample standard deviation s = 0.76 years. However, it isthought that the overall population mean age of coyotes is ? = 1.75. Do the sample dataindicate that coyotes in this region of northern Minnesota tend to live longer than theaverage of 1.75 years? Use ? = 0.01.(a) What is the level of significance?State the null and alternate hypotheses. belowH0: ? = 1.75 yr; H1: ? < 1.75 yrH0: ? = 1.75 yr; H1: ? > 1.75 yrH0: ? < 1.75 yr; H1: ? = 1.75 yrH0: ? = 1.75 yr; H1: ? ? 1.75 yrH0: ? > 1.75 yr; H1: ? = 1.75 yr(b) What sampling distribution will you use? Explain the rationale for your choiceof sampling distribution.The standard normal, since the sample size is large and ? is unknown.The standard normal, since the sample size is large and ? is known.The Student’s t, since the sample size is large and ? is known.The Student’s t, since the sample size is large and ? is unknown. What is the value of the sample test statistic? (Round your answer to threedecimal places.)(c) Find the P-value. (Round your answer to four decimal places.) 10. Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of”good,” socially conscious companies are Johnson and Johnson, Dell Computers, Bank ofAmerica, and Home Depot. The question is, are such stocks overpriced? One measure ofvalue is the P/E, or price-to-earnings ratio. High P/E ratios may indicate a stock is overpriced.For the S&P Stock Index of all major stocks, the mean P/E ratio is ? = 19.4. A randomsample of 36 “socially conscious” stocks gave a P/E ratio sample mean of x = 18.5, withsample standard deviation s = 4.8. Does this indicate that the mean P/E ratio of all sociallyconscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index?Use ? = 0.10.(a) What is the level of significance?State the null and alternate hypotheses.H0:H0:H0:H0:H0: ? = 19.4; H1: ? < 19.4? = 19.4; H1: ? > 19.4? > 19.4; H1: ? = 19.4? = 19.4; H1: ? ? 19.4? ? 19.4; H1: ? = 19.4 (b) What sampling distribution will you use? Explain the rationale for your choiceof sampling distribution.The Student’s t, since the sample size is large and ? is unknown.The standard normal, since the sample size is large and ? is unknown.The Student’s t, since the sample size is large and ? is known.The standard normal, since the sample size is large and ? is known.What is the value of the sample test statistic? (Round your answer to threedecimal places.)(c) Find the P-value. (Round your answer to four decimal places.) 11. The Student’s t distribution table gives critical values for the Student’s t distribution. Use anappropriate d.f. as the row header. For a right-tailed test, the column header is the valueof ? found in the one-tail area row. For a left-tailed test, the column header is the valueof ? found in the one-tail area row, but you must change the sign of the critical value t to ?t. Fora two-tailed test, the column header is the value of ? from the two-tail area row. The criticalvalues are the ±t values shown. Pyramid Lake is on the Paiute Indian Reservation in Nevada. The lake is famous for cutthroattrout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is ? = 19 inches. However, a survey reported that of a random sample of 51 fish caught, the meanlength was x = 18.4 inches, with estimated standard deviation s = 2.6 inches. Do these dataindicate that the average length of a trout caught in Pyramid Lake is less than ? = 19 inches?Use ? = 0.05. Solve the problem using the critical region method of testing (i.e., traditionalmethod). (Round the your answers to three decimal places.)test statistic =critical value = State your conclusion in the context of the application.Reject the null hypothesis, there is sufficient evidence that the average fish length is less than 19inches.Reject the null hypothesis, there is insufficient evidence that the average fish length is less than 19inches.Fail to reject the null hypothesis, there is sufficient evidence that the average fish length is lessthan 19 inches.Fail to reject the null hypothesis, there is insufficient evidence that the average fish length is lessthan 19 inches….pare your conclusion with the conclusion obtained by using the P-value method. Arethey the same?The conclusions obtained by using both methods are the same.We reject the null hypothesis using the traditional method, but fail to reject using the P-valuemethod.We reject the null hypothesis using the P-value method, but fail to reject using the traditionalmethod.