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Hypothesis Testing of Single Mean and Single Proportion: Using the Sample to Support One of the Hypotheses

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

Use the sample (data) to calculate the actual significance of the test or the p-value. The p-value is the probability that an outcome of the data (for example, the sample mean) will happen purely by chance when the null hypothesis is true.

A large p-value calculated from the data indicates that the outcome of the data is happening purely by chance. The data supports the null hypothesis so we do not reject it. The smaller the p-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against the null hypothesis.

The p-value is sometimes called the computed αα because it is calculated from the data. You can think of it as the probability of (incorrectly) rejecting the null hypothesis when the null hypothesis is actually true.

Draw a graph that shows the p-value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.

Example 1: (to illustrate the p-value)

Suppose a baker claims that his bread height is more than 15 cm, on the average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The average height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm.

The null hypothesis could be HoHo: μ15μ15 The alternate hypothesis is HaHa: μ>15μ>15

The words "is more than" translates as a ">>" so "μ>15μ>15" goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.

Since σσ is known (σ=0.5σ=0.5 cm.), the distribution for the test is normal with mean μμ=15=15 and standard deviation σnσn =0.510=0.16=0.510=0.16.

Suppose the null hypothesis is true (the average height of the loaves is no more than 15 cm). Then is the average height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample average would be if the null hypothesis were true. The graph shows how far out the sample average is on the normal curve. How far out the sample average is on the normal curve is measured by the p-value. The p-value is the probability that, if we were to take other samples, any other sample average would fall at least as far out as 17 cm.

The p-value, then, is the probability that a sample average is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for averages from Chapter 7.

Normal distribution curve on average bread heights with values 15, as the population mean, and 17, as the point to determine the p-value, on the x-axis.

p-value=P(X¯>17)p-value=P(X>17) which is approximately 0.

A p-value of approximately 0 tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on the average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE. Because the outcome of 17 cm. is so unlikely (meaning it is happening NOT by chance but is a rare event), we conclude that the evidence is strongly against the null hypothesis (the average height is at most 15 cm.). There is sufficient evidence that the true average height for the population of the baker's loaves of bread is greater than 15 cm.


A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called null hypothesis (notation H0H0 size 12{H rSub { size 8{0} } } {}) and contradictory statement is called alternate hypothesis (notation HaHa size 12{H rSub { size 8{a} } } {}).
The probability that event will happen purely by chance assuming the null hypothesis is true. The smaller p-value, the stronger the evidence is against the null hypothesis.
Standard Deviation:
A number that is equal to the square root of the variance and measures how far data values are from their mean. Notations: s for sample standard deviation and σσfor population standard deviation.

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