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# Hypothesis Testing of Single Mean and Single Proportion: Examples

Summary: This module provides examples of Hypothesis Testing of a Single Mean and a Single Proportion as a part of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

## Example 1

### Problem 1

Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds. His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster by using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims. For the 15 swims, Jeffrey's mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α=0.05α=0.05. Assume that the swim times for the 25-yard freestyle are normal.

Historical Note: The traditional way to compare the two probabilities, αα and the p-value, is to compare the critical value (z-score from αα) to the test statistic (z-score from data). The calculated test statistic for the p-value is -2.08-2.08. (From the Central Limit Theorem, the test statistic formula is z = x¯ - μ X ( σ X n ) z= x - μ X ( σ X n ) . For this problem, x¯=16 x =16, μ X =16.43 μ X =16.43 from the null hypothesis, σ X =0.8 σ X =0.8, and n =15n=15.) You can find the critical value for α=0.05α=0.05 in the normal table (see 15.Tables in the Table of Contents). The z-score for an area to the left equal to 0.05 is midway between -1.65 and -1.64 (0.05 is midway between 0.0505 and 0.0495). The z-score is -1.645. Since -1.645>-2.08-1.645>-2.08 (which demonstrates that α>p-valueα>p-value), reject HoHo. Traditionally, the decision to reject or not reject was done in this way. Today, comparing the two probabilities αα and the p-value is very common. For this problem, the p-value, 0.01870.0187 is considerably smaller than αα, 0.05. You can be confident about your decision to reject. The graph shows αα, the p-value, and the test statistics and the critical value.

## Example 2

### Problem 1

A college football coach thought that his players could bench press a mean weight of 275 pounds. It is known that the standard deviation is 55 pounds. Three of his players thought that the mean weight was more than that amount. They asked 30 of their teammates for their estimated maximum lift on the bench press exercise. The data ranged from 205 pounds to 385 pounds. The actual different weights were (frequencies are in parentheses) 205(3); 215(3); 225(1); 241(2); 252(2); 265(2); 275(2); 313(2); 316(5); 338(2); 341(1); 345(2); 368(2); 385(1). (Source: data from Reuben Davis, Kraig Evans, and Scott Gunderson.)

Conduct a hypothesis test using a 2.5% level of significance to determine if the bench press mean is more than 275 pounds.

## Example 3

### Problem 1

Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores 65; 65; 70; 67; 66; 63; 63; 68; 72; 71. He performs a hypothesis test using a 5% level of significance. The data are from a normal distribution.

## Example 4

### Problem 1

Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50%. Joon samples 100 first-time brides and 53 reply that they are younger than their grooms. For the hypothesis test, she uses a 1% level of significance.

## Example 5

### Problem 1

Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey 150 households with the result that 43 of the households have three cell phones.

The next example is a poem written by a statistics student named Nicole Hart. The solution to the problem follows the poem. Notice that the hypothesis test is for a single population proportion. This means that the null and alternate hypotheses use the parameter pp. The distribution for the test is normal. The estimated proportion p'p' is the proportion of fleas killed to the total fleas found on Fido. This is sample information. The problem gives a preconceived α=0.01α=0.01, for comparison, and a 95% confidence interval computation. The poem is clever and humorous, so please enjoy it!

## Note:

Hypothesis testing problems consist of multiple steps. To help you do the problems, solution sheets are provided for your use. Look in the Table of Contents Appendix for the topic "Solution Sheets." If you like, use copies of the appropriate solution sheet for homework problems.

## Example 6

### Problem 1

My dog has so many fleas,
They do not come off with ease.
As for shampoo, I have tried many types
Even one called Bubble Hype,
Which only killed 25% of the fleas,

I've used all kinds of soap,
Until I had give up hope
Until one day I saw
An ad that put me in awe.

A shampoo used for dogs
Called GOOD ENOUGH to Clean a Hog
Guaranteed to kill more fleas.

I gave Fido a bath
And after doing the math
His number of fleas
Started dropping by 3's!

Before his shampoo
I counted 42.
At the end of his bath,
I redid the math
And the new shampoo had killed 17 fleas.

Now it is time for you to have some fun
With the level of significance being .01,
You must help me figure out
Use the new shampoo or go without?

## Glossary

Central Limit Theorem:
Given a random variable (RV) with known mean μμ and known standard deviation σσ. We are sampling with size n and we are interested in two new RVs - the sample mean, X¯X¯, and the sample sum, ΣXΣX. If the size nn of the sample is sufficiently large, then X¯X¯ size 12{ { bar {X}}} {} N μ σ n N μ σ n and ΣXΣX size 12{X} {}N ( , n σ )N(,nσ). If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n σ n , is called the standard error of the mean.
Standard Deviation:
A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and σσ for population standard deviation.

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