Question

1) Assuming that both use the same sample standard deviation and confidence level, why are confidence intervals from Excel’s CONFIDENCE.NORM function narrower than those from CONFIDENCE.T? Which of the two is better and why?

2) Define p-value.

3) Suppose you look at your output for a test of a single mean and see a t-statistic of -2.5. In a single sentence, describe what the number -2.5 represents.

4) Why would one prefer to have the most serious potential error in hypothesis testing be a Type I error?

Answer 1

Solution1:

CONFIDENCE.NORM is used when population standard deviation is known and uses Z critical values

and n>30

and CONFIDENCE.T is used when  population standard deviation is not known and uses t critical and n values are less than 30.

1. The t-distribution incorporates the fact that for smaller sample sizes the distribution will be more spread out using something called degrees of freedom. For confidence intervals, the degrees of freedom will allways be df=n?1, or one less than the sample size.
2. For every change in degrees of freedom, the t-distribution changes. The larger the sample size (n), the closer the t-distribution mimics the z-distribution in shape. We construct a confidence interval for a small sample size in the same way as we do for a large sample, except we use the t-distribution instead of the z-distribution.

For ex

=CONFIDENCE.NORM(0.05;5;30)

=1.789194

=CONFIDENCE.T(0.05;5;30)

=1.867031

as sample szie gets larger n>30 ,t distributon becomes z distribution.

df=n-1

as as df changes t crit changes ,

but for n>30,z crit will be same for given alpha

Hence,CONFIDENCE.NORM function narrower than those from CONFIDENCE.T

Solution2:

2) Define p-value.

Assuming null hypothesis is true,the P-value is the probability of observing a sample mean that is as or more extreme than the observed.

calcualte p and compare to alpha

if p<alpha, the results are statistically significant that is unlikely due to chance

ifp>alpha, the results are not significant and they are due to chance.

3) Suppose you look at your output for a test of a single mean and see a t-statistic of -2.5. In a single sentence, describe what the number -2.5 represents.

t=-2.5

that is it is 2.5 standard deviations below mean