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 *n*_{1} = 9
children (9 years old) showed that they had an average REM sleep
time of *x*_{1} = 2.9 hours per night. From previous
studies, it is known that *σ*_{1} = 0.6 hour.
Another random sample of *n*_{2} = 9 adults showed
that they had an average REM sleep time of *x*_{2} =
2.40 hours per night. Previous studies show that
*σ*_{2} = 0.7 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 *μ*_{1} − *μ*_{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

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?

The conclusions obtained by using both methods are the
same.These two methods differ slightly. 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.

Answer #1

x_{1}
= |
2.90 |
x_{2}
= |
2.40 |

n_{1}
= |
9 |
n_{2}
= |
9 |

σ_{1}
= |
0.60 |
σ_{2}
= |
0.70 |

std
error
σ_{}_{1-}_{}_{2}=√(σ^{2}1/n_{1}+σ^{2}_{2}/n_{2})
= |
0.307 | ||

test stat z
=(x_{1}-x_{2}-Δ_{o})/σ_{}_{1-}_{}_{2 }
= |
1.63 |

for 0.1 level with right tail test , critical
z= |
1.28 |

p value : = |
0.0516 |

Conclusion:

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

The conclusions obtained by using both methods are the same

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...

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 = 10
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...

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...

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 = 2.6 hours per night. From previous
studies, it is known...

(11) 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 = 2.6 hours per night. From
previous studies, it is...

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 = 10 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...

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 = 10 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...

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 = 2.5 hours per night. From previous
studies, it is known...

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...

REM sleep is when most people dream. Each night, adults have
both REM and non-REM sleep. It is thought that children have more
REM sleep than adults. A random sample of 10 children had an
average REM sleep time of 2.8 hours. A random sample of 10 adults
had an average REM sleep time of 2.1 hours. From previous studies,
the standard deviations are known to be 0.5 hour for the children’s
group and 0.7 hour for the adult group....

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