Test of hypothesis (two-tail)

We shall be testing hypotheses about the population mean (it is assumed we know the variance of the population). We shall fail to reject the hypotesis if the obseved sample mean (x-bar) is close to the hypothesized population mean (µ). We will shall reject the hypothesis if the observed sample mean (x-bar) is far from the hypothesized population mean (µ). Distance will be measured in standard deviation units (z-scores) based on the hypothesis being true, or the probability of the z-score being that large.

A test of hypotheis entails:

Calculating the z-score.
(Comparing the z-score to the critical value, which we will not do, or) coverting the z-score to a p-value.
Comparing the p-value to the significance level.

Example: If you are told that the mean weight of 3rd graders is 85 pounds with a standard deviation of 20 pounds, and you find that the mean weight of a group of 22 students is 95 pounds, do you question that that group of students is a group of third graders?

The z-score is ((x-bar) - µ)/(*sigma*/(n^.5)); the numerator is the difference between the observed and hypothesized mean, the denominator rescales the unit of measurement to standard deviation units. (95-85)/(20/(22^.5)) = 2.3452.
The z-score 2.35 corresponds to the probability .9906, which leaves .0094 in the tail beyond. Since one could have been as far below 85, the probability of such a large or larger z-score is .0188. This is the p-value. Note that for these two tailed tests we are using the absolute value of the z-score.
Because .0188 < .05, we reject the hypothesis (which we shall call the null hypothesis) at the 5% significance level; if the null hypothesis were true, we would get such a large z-score less than 5% of the time. Because .0188 > .01, we fail to reject the null hypothesis at the 1% level; if the null hypothesis were true, we would get such a large z-score more than 1% of the time.

N.B.:

You reject the null hypothesis if the z-score is large, which means that the p-value is small.
If you reject a hypothesis at the 5% significance level, p < .05, hence you will reject that hypothesis at the 10% significance level.
If you fail to reject a hypothesis at the 5% significance level, p > .05, hence you will fail to reject that hypothesis at the 1% significance level.
Competencies: If the standard deviation is known to be equal to 12, and your null hypothesis is that the mean of the population is equal to 15, at what level (p-value) is x-bar = 13.5 based on a sample of size 200 significant? Would you reject the null hypothesis at the 10% significance level? 5% significance level? 1% significance level?
Reflection: What is the relationship between a two tailed test of hypothesis and a confidence interval?
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Questions?