CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 8. Hypothesis Testing

Remember that since we have a random sample from the population, we do not expect the sample mean to beexactly
equal to the hypothesized value of the population mean. Therefore, the question really is: “How different can the
observed sample mean be from the hypothesized mean before rejecting the null hypothesis?” Or, in other words, “If
the null hypothesis is true, is it likely that we will obtain such an observed sample mean?” We use our critical values
taken from thez-distribution to determine those cutoffs.

To evaluate the sample mean against the hypothesized population mean, we use the concept ofz-scores to determine
how different the two means are from each other. As we learned in previous lessons, thez-score is calculated by
using the formula:

z=

(X ̄−μ)

σX

where:

z=standardized score

X ̄=sample mean

μ=hypothesized population mean

σX=standard error?. If we do not have the population variance, we can estimate the deviation of the samples
from the true population mean by dividing the standard deviation by the square root of the number of observations(

√σ

n

)

.

Once we calculate thez-score, we can make a decision about whether to reject or to fail to reject the null hypothesis
based on the critical values.

Let’s calculate the test statistic for several different scenarios.

Example:

College A has an average SAT score of 1, 500 .From a random sample of 125 freshman psychology students we find
the average SAT score to be 1,450 with a standard deviation of 100.We want to know if these freshman psychology
students are representative of the overall population. What are our hypotheses and the test statistic?

Solution:

Let’s first develop our null and alternative hypotheses:

H 0 :μ= 1500

Ha:μ 6 = 1500

Our standardz-score for the sample of freshman psychology students would be:

z=

X ̄−μ

σx

=

1450 − 1500

100 /

√

125

≈− 5. 59

Example:

A farmer is trying out a planting technique that he hopes will increase the yield on his pea plants. Over the last
5 years,the average number of pods on one of his pea plants was 145 pods with a standard deviation of 100 pods.This
year, after trying his new planting technique, he takes a random sample of his plants and finds the average number
of pods to be 147.He wonders whether or not this is a statistically significant increase. What is his hypotheses and
the test statistic?