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 adultsâ€ .

Assume that REM sleep time is normally distributed for both

children and adults. A random sample of *n*_{1} = 10

children (9 years old) showed that they had an average REM sleep

time of *x*_{1} = 2.6 hours per night. From previous

studies, it is known that *Ïƒ*_{1} = 0.6 hour.

Another random sample of *n*_{2} = 10 adults showed

that they had an average REM sleep time of *x*_{2} =

1.90 hours per night. Previous studies show that

*Ïƒ*_{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.

(a) What is the level of significance?

State the null and alternate hypotheses.

*H*_{0}: *Î¼*_{1} =

*Î¼*_{2}; *H*_{1}:

*Î¼*_{1} >

*Î¼*_{2}*H*_{0}:

*Î¼*_{1} = *Î¼*_{2};

*H*_{1}: *Î¼*_{1} â‰

*Î¼*_{2} *H*_{0}:

*Î¼*_{1} < *Î¼*_{2};

*H*_{1}: *Î¼*_{1} =

*Î¼*_{2}*H*_{0}:

*Î¼*_{1} = *Î¼*_{2};

*H*_{1}: *Î¼*_{1} <

*Î¼*_{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 known standard

deviations.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.

What is the value of the sample test statistic? (Test the

difference *Î¼*_{1} âˆ’ *Î¼*_{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 *Î±*?

At the *Î±* = 0.01 level, we reject the null hypothesis

and conclude the data are statistically significant.At the

*Î±* = 0.01 level, we fail to reject the null hypothesis and

conclude the data are not statistically

significant. At the *Î±* = 0.01

level, we reject the null hypothesis and conclude the data are not

statistically significant.At the *Î±* = 0.01 level, we fail

to 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 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. 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.