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 = 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 = 0.8 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 1.80 hours per night. Previous studies show that 𝜎2 = 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. What is the value of the sample test statistic? (Test the difference 𝜇1 − 𝜇2. Round your answer to two decimal places.) Find (or estimate) the P-value. (Round your answer to four decimal places.)

          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 = 0.8 hour. Another random
sample of n2 = 9 adults showed that they had an
average REM sleep time of x2 = 1.80 hours per
night. Previous studies show that 𝜎2 = 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.
What is the value of the sample test statistic? (Test the
difference 𝜇1 − 𝜇2. Round your answer to two
decimal places.)
 Find (or estimate) the P-value. (Round your answer
to four decimal places.)

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Sri K

07/31/2023

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The alternative hypothesis is that children have more REM sleep time than adults. Show more…

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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 = 0.8 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 1.80 hours per night. Previous studies show that 𝜎2 = 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. What is the value of the sample test statistic? (Test the difference 𝜇1 − 𝜇2. Round your answer to two decimal places.) Find (or estimate) the P-value. (Round your answer to four decimal places.)

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00:01 Here given n1 it is equal to n1 is equal to 9 children's and n2 is equal to 9 adults, n2 is equal to 9 adults.

00:16 So, x1 it is nothing but the sleep time of x1 it is nothing but 2 .6 hours per night and sigma1 it is nothing but 0 .9 hours.

00:30 Whereas in terms of x2 it is 1 .70 and sigma2 it is nothing but 0 .6 hours.

00:44 So using 1 % level of significance we have to calculate the concept.

00:49 So, 1 % level of significance it is nothing but 0 .01, 0 .01.

00:58 So, here let us consider our null hypothesis h naught to be null hypothesis h naught to be mu1 is equal to mu2 and the alternative hypothesis h alpha or h1 to be mu1 greater than mu2, mu1 greater than mu2.

01:19 So, now based upon the given data that is we have to test if these data indicate that on average children tend to have more rem sleep than adult or not.

01:33 So, this will be the concept.

01:35 For this purpose we are going to use the formula z is equal to x1 bar minus x2 bar divided by square root of sigma1 square divided by n1 plus sigma2 square divided by n2.

01:52 So, here all the values are given already that is x1, x2 that is 2 .6 and 2 .6 minus 1 .7 divided by square root of sigma1 square 0 .9 and sigma2 square 0 .6.

02:10 So, it will be in terms of 0 .9 square divided by 9 n1 plus 0 .6 the whole square divided by 9 which is nothing but n2.

02:22 So, on simplifying this process we will be obtaining the solution to be 1 .19...

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