Question

In a​ study, researchers wanted to measure the effect of alcohol on the hippocampal​ region, the portion of the brain responsible for​ long-term memory​ storage, in adolescents. The researchers randomly selected 10 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm cubed. An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and x overbarequals8.11 cm cubed and sequals0.8 cm cubed. Conduct the appropriate test at the alpha=0.01 level of significance.

State the null and alternative hypotheses.

Upper H 0​: ​(Type integers or decimals. Do not​ round.)

Identify the​ t-statistic. ​(Round to two decimal places as​ needed.)

Identify the​ P-value. ​(Round to three decimal places as​ needed.)

Make a conclusion regarding the hypothesis.

Answer 1

Solution:

We have to test  whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm cubed.

Sample size = n = 10

Sample mean =

Sample Standard deviation = s = 0.8

level of significance = 0.01

Step 1) State the null and alternative hypotheses.

vs   ( this is left tailed test)

Step 2) Identify the​ t-statistic. ​

Step 3) P-value:

Use following steps in TI 84:

Press STAT and select TESTS

Under TESTS, select TTest

Select Stats

Under Stats , enter numbers

Click on Calculate and press Enter

Here we get both t test statistic value as well as P-value:

p = 0.002887 = 0.003

thus

P-value = 0.003

We can excel also:

=T.DIST(t,df,cumulative)

where df = n -1 = 10 - 1 = 9

thus

=T.DIST(-3.60,9,TRUE)

=0.0028742

=0.003

Thus P-value = 0.003

Step 4) Make a conclusion regarding the hypothesis.

Decision Rule:
Reject null hypothesis H0, if P-value < 0.01 level of significance, otherwise we fail to reject H0

Since P-value = 0.003 < 0.01 level of significance, we reject null hypothesis H0.

Thus at 0.01 level of significance, we have sufficient evidence to conclude that: the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm cubed.