Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 9 children (9 years old
) showed that they had an average REM sleep time of x1 = 2.9 hours per night. From previous studies, it is known that σ1 = 0.5 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.2 hours per night. Previous studies show that σ2 = 0.7 hour.(a) Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.(i) What is the level of significance?State the null and alternate hypotheses.A) H0: μ1 = μ2; H1: μ1 < μ2B) H0: μ1 = μ2; H1: μ1 > μ2 C) H0: μ1 > μ2; H1: μ1 = μ2D) H0: μ1 = μ2; H1: μ1 ≠ μ2E) H0: μ1 < μ2; H1: μ1 = μ2(ii) What sampling distribution will you use? What assumptions are you making?A) The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.B) The standard normal. We assume that both population distributions are approximately normal with known standard deviations. C) The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.D) The Student's t. We assume that both population distributions are approximately normal with known standard deviations.What is the value of the sample test statistic? (iii) Find the P-value.(iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?A) Since the P-value > α, we fail to reject the null hypothesis. The data are not statistically significant.B) Since the P-value > α, we reject the null hypothesis. The data are not statistically significant. C) Since the P-value ≤ α, we reject the null hypothesis. The data are statistically significant.D) Since the P-value ≤ α, we fail to reject the null hypothesis. The data are statistically significant.(v) Interpret your conclusion in the context of the application.A) Reject H0. At the 1% level of significance, the evidence is sufficient to indicate that the population mean REM sleep time for children is more than for adults.B) Fail to reject H0. At the 1% level of significance, the evidence is sufficient to indicate that the population mean REM sleep time for children is less than for adults. C) Reject H0. At the 1% level of significance, the evidence is insufficient to indicate that the population mean REM sleep time for children is more than for adults.D) Fail to reject H0. At the 1% level of significance, the evidence is insufficient to indicate that the population mean REM sleep time for children is less than for adults.(b) Find a 98% confidence interval for μ1 − μ2. Explain the meaning of the confidence interval in the context of the problem. A) At the 98% level of confidence, it appears that the population mean REM sleep time for children is greater than that for adults.B) At the 98% level of confidence, it appears that the population mean REM sleep time for children is less than that for adults. C) At the 98% level of confidence, it appears that the population mean REM sleep time for children is not different from that for adults.
By Stokes' theorem, the line integral of over is given by the surface integral of the curl of over , where is the region of intersection of the plane and the cylinder with having positive/upward orientation.
