(iv) Based on your answers in parts (1)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? O At the a = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. O At the a= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. O At the a= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. O At the a= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. (v) Interpret your conclusion in the context of the application. O Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. O Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. ) Find a 98% confidence interval for ₁- ₂- (Round your answers to two decimal places.) lower limit upper limit Explain the meaning of the confidence interval in the context of the problem. O Because the interval contains only positive numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is greater than that for adults. O Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we can not say that the population mean REM sleep time for children is greater than that for adults. Ⓒ We can not make any conclusions using this confidence interval. O Because the interval contains only negative numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is less than that for adults.

(iv) Based on your answers in parts (1)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? O At the a = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. O At the a= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. O At the a= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. O At the a= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. (v) Interpret your conclusion in the context of the application. O Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. O Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. ) Find a 98% confidence interval for ₁- ₂- (Round your answers to two decimal places.) lower limit upper limit Explain the meaning of the confidence interval in the context of the problem. O Because the interval contains only positive numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is greater than that for adults. O Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we can not say that the population mean REM sleep time for children is greater than that for adults. Ⓒ We can not make any conclusions using this confidence interval. O Because the interval contains only negative numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is less than that for adults.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults.†

Assume that REM sleep time is normally distributed for both children and adults. A random sample of n₁ = 11 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 ?₁ = 0.6 hour. Another random sample of n₂ = 11 adults showed that they had an average REM sleep time of x₂ = 2.2 hours per night. Previous studies show that ?₂ = 0.7 hour.

(a)

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults.
Assume that REM sleep time is normally distributed for both children and adults. A random sample of n, 11 children (9 years old) showed that they had an average REM sleep time of x₂ = 2.9 hours per night. From previous studies, it is known that a, 0.6 hour. Another random sample of n₂ = 11 adults.
showed that they had an average REM sleep time of x₂ = 2.2 hours per night. Previous studies show that ₂ -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.
(1) What is the level of significance?
State the null and alternate hypotheses.
H₂i Hqi Hy > H₂
Ho: M₁
ⒸH₂ H₁ H₂i H₂= H₂ <H₂
ⒸH₂² M₂ <H₂i H₂ M₂ M₂
O Hoi H₂H₂i H₁ H₂ H₂
(ii) What sampling distribution will you use? What assumptions are you making?
O The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
O The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
O The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
What is the value of the sample test statistic? Compute the corresponding z or t-value as appropriate. (Test the difference ₂-₂. Round your answer to two decimal places.)
(iii) Find (or estimate) the P-value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-value.
0-3
0
u
70-3
-1
0
2
60-3
-2 -1
(iv) Based on your answers in parts (1)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a?
At the a= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
O At the a
0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
O At the
0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
O At the a= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
(v) Interpret your conclusion in the context of the application.
O Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
O Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
O Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
O Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
0
2
^
-2
-1
0
20-
'D

(iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant
O At the a = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
O At the a = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the a = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
O At the a= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
level a?
(v) Interpret your conclusion in the context of the application.
O Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
O Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
O Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
O Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
(b) Find a 98% confidence interval for #₁ #₂. (Round your answers to two decimal places.)
lower limit
upper limit
Explain the meaning of the confidence interval in the context of the problem.
O Because the interval contains only positive numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is greater than that for adults.
O Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we can not say that the population mean REM sleep time for children is greater than that for adults.
O We can not make any conclusions using this confidence interval.
O Because the interval contains only negative numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is less than that for adults.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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