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## Performing a One-sample Z-test

1. Devise a null and alternative hypothesis, then perform a one sample z-test using alpha=0.05 if our class is taller or shorter than the average human in the United States. Graphically display your answer and statistically write up results. In addition, determine the effect size of this relationship and interpret.
Notes:
1. The average height of adults in the United States is 67.0±3.8 inches. (Note: this is a population SD)
2. Calculate and interpret the effect size (use Cohen’s d):
Cohen's d = (Msample - µpopulation) ⁄ σ)
Ans: The one-sample z-test hypothesis is specified as follows: The test statistic z is calculated to be 1.59. The cutoff value for two-sided z-test at 5% level of significance is 1.96. Since , we failed to reject out null hypothesis in favor of alternative.
 Mean 68.26 Population Mean 67.00 Population SD 3.80 N 23.00 Z 1.59 Cut-off 1.96 p-value 0.11

We conclude that there is not enough evidence in the sample data to claim that our class is taller or shorter than the average human in the United States.

We represent the sample data by a plot below:  The Cohen’s effect size is

The effect size of the sample is 0.33 which is also small, so we do not expect the difference to be too significant even in case the sample was very large.

## Performing a T-test

Devise a null and alternative hypothesis, then perform a t-test using alpha =0.05 if the students in our class that are born in warm weather months are taller or shorter than those born in cold weather months. Graphically display your answer and statistically write up results. In addition, determine the effect size of this relationship and interpret.

Notes:

Population SDs are unknown.

Assume the northern hemisphere:

Warm months = April, May, June, July, August, September

Cold months = October, November, December, January, February, March

Use Cohen’s d effect size:

Cohen’s d = (M2 - M1) ⁄ SDpooled) where SDpooled = √((SD12 + SD22) ⁄ 2)

Ans:

The two-sample t-test hypothesis is specified as follows: We have considered Mar-Aug as warm weather (Sample 1) and Sep-Feb (Sample 2) as cold weather. We have assumed that the variance in both groups is same and hence, we would use the pooled variance to calculate two-sample t-test.

SDpooled = √((SD12 + SD22) ⁄ 2)

The test statistic t is calculated to be 0.227. The cutoff value for two-sided t-test at 5% level of significance is 2.08 for 21 degrees of freedom. Since , we failed to reject out null hypothesis in favor of alternative.

 Mean of Warm Weather 68.45 Mean of Cold Weather 68.08 Diff tested 0 SD Warm Weather 2.11 SD Cold Weather 5 Pooled variance 3.9 T-stat 0.227 Two-Sided Cut-off 2.08
The test statistic t is calculated to be 0.227. The cutoff value for two-sided t-test at 5% level of significance is 2.08 for 21 degrees of freedom. Since , we failed to reject out null hypothesis in favor of alternative.  The effect size is The effect size of the sample is 0.95 which is very small, so we do not expect the difference to be too significant even in case the sample was very large.