# REM (rapid eye movement) sleep

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 = 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 ?1 = 0.8 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.10 hours per night. Previous studies show that ?2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H0: ?1 = ?2H1: ?1 < ?2
H0: ?1 = ?2H1: ?1 ≠ ?2
H0: ?1 = ?2H1: ?1 > ?2
H0: ?1 < ?2H1: ?1 = ?2

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student’s t. We assume that both population distributions are approximately normal with known standard deviations.
The Student’s t. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference ?1 − ?2. Round your answer to two decimal places.)

(c) 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.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

At the ? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the ? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

# REM (rapid eye movement) sleep

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 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.6 hours per night. From previous studies, it is known that σ1 = 0.6 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 1.90 hours per night. Previous studies show that σ2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H0: μ1 < μ2H1: μ1 = μ2H0: μ1 = μ2H1: μ1 > μ2    H0: μ1 = μ2H1: μ1 ≠ μ2H0: μ1 = μ2H1: μ1 < μ2

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.The Student’s t. We assume that both population distributions are approximately normal with unknown standard deviations.    The Student’s t. We assume that both population distributions are approximately normal with known standard deviations.The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

# REM (rapid eye movement) sleep

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 = 2.7 hours per night. From previous studies, it is known that ?1 = 0.9 hour. Another random sample of n2 = 11 adults showed that they had an average REM sleep time of x2 = 2.20 hours per night. Previous studies show that ?2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference ?1 − ?2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

 test statistic critical value P-value

Conclusion

Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.    Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.    The conclusions obtained by using both methods are the same.These two methods differ slightly.

# REM (rapid eye movement) sleep

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 = 9 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.7 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that ?2 = 0.8 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference ?1 − ?2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

 test statistic critical value P-value

# REM (rapid eye movement) sleep

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 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.6 hours per night. From previous studies, it is known that σ1 = 0.6 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 1.90 hours per night. Previous studies show that σ2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

# REM (rapid eye movement) sleep

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 = 8 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 = 0.9 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that ?2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H0: ?1 = ?2H1: ?1 ≠ ?2H0: ?1 = ?2H1: ?1 < ?2    H0: ?1 = ?2H1: ?1 > ?2H0: ?1 < ?2H1: ?1 = ?2

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.    The Student’s t. We assume that both population distributions are approximately normal with unknown standard deviations.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? (Test the difference ?1 − ?2. Round your answer to two decimal places.)

(c) 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.

# REM (rapid eye movement) sleep

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 = 9 children (9 years old) showed that they had an average REM sleep time of x1 = 2.7 hours per night. From previous studies, it is known that σ1 = 0.7 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.20 hours per night. Previous studies show that σ2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference μ1 − μ2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

 test statistic critical value P-value

# REM (rapid eye movement) sleep

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 = 2.5 hours per night. From previous studies, it is known that σ1 = 0.7 hour. Another random sample of n2 = 11 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that σ2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference μ1 − μ2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

 test statistic critical value P-value

Conclusion

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.    Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.These two methods differ slightly.    We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.The conclusions obtained by using both methods are the same.

# REM (rapid eye movement) sleep

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 = 2.20 hours per night. Previous studies show that ?2 = 0.7 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

State the null and alternate hypotheses.

H0: ?1 = ?2H1: ?1 ≠ ?2
H0: ?1 < ?2H1: ?1 = ?2
H0: ?1 = ?2H1: ?1 > ?2
H0: ?1 = ?2H1: ?1 < ?2

(b) What sampling distribution will you use? What assumptions are you making?

The Student’s t. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The Student’s t. We assume that both population distributions are approximately normal with known standard deviations.

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? (Test the difference ?1 − ?2. Round your answer to two decimal places.)

Find (or estimate) the P-value. (Round your answer to four decimal places.)

Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

At the ? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the ? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

# REM (rapid eye movement) sleep

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 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.6 hours per night. From previous studies, it is known that σ1 = 0.6 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 1.90 hours per night. Previous studies show that σ2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to two decimal places.)

# REM (rapid eye movement) sleep

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 = 9 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.6 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.10 hours per night. Previous studies show that ?2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference ?1 − ?2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

 test statistic critical value P-value

Conclusion

A. Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
B. Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
C. Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
D. Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

A. We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.
B. We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.
C. The conclusions obtained by using both methods are the same.
D. These two methods differ slightly.

# REM (rapid eye movement) sleep

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 σ1 = 0.8 hour. Another random sample of n2 = 11 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that σ2 = 0.9 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to two decimal places.)

State the null and alternate hypotheses.

H0: μ1 = μ2H1: μ1 < μ2
H0: μ1 < μ2H1: μ1 = μ2
H0: μ1 = μ2H1: μ1 > μ2
H0: μ1 = μ2H1: μ1 ≠ μ2
(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student’s t. We assume that both population distributions are approximately normal with unknown standard deviations.
The Student’s t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

(c) 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.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

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