**Hypothesis Testing of Single Mean**

**Abstract**

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

**Example 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 smean 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. Assume that he swim times for the 25-yard freestyle are normal.

**Solution**

Set up the Hypothesis Test:

Since the problem is about a mean, this is a test of a single population mean.

**Graph:**

** **

The Type I and Type II errors for this problem are as follows:

The Type I error is to conclude that Jeffrey swims the 25-yard freestyle, on average, in less than

16.43 seconds when, in fact, he actually swims the 25-yard freestyle, on average, in 16.43 seconds.

(Reject the null hypothesis when the null hypothesis is true.)

The Type II error is that there is no evidence to conclude that Jeffrey swims the 25-yard

freestyle, on average, in less than 16.43 seconds when, in fact, he actually does swim the 25-yard freestyle, on average, in less than 16.43 seconds. (Do not reject the null hypothesis when the null-hypothesis is false.)

**Example 2**

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

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 obtain 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

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 replies that they are younger than their grooms.

For the hypothesis test, she uses a 1% level of significance.

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 “SolutionSheets.” If you like, use copies of the appropriate solution sheet for homework problems.

Example 6

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,

Unfortunately, I was not pleased.

I’ve used all kinds of soap,

Until I had given 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.

So now I was pleased.

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?

Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence

that the percentage of fleas that are killed by the new shampoo is more than 25%.

Construct a 95% Confidence Interval for the true mean or proportion. Include a sketch of the

graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval.

Confidence Interval: (0:26; 0:55) We are 95% confident that the true population proportion

POF fleas that are killed by the new shampoo is between 26% and 55%.

note: This test result is not very definitive since the p-value is very close to alpha. In reality, one

would probably do more tests by giving the dog another bath after the fleas have had a chance to

return.

**Solution**** **

**1**.

**One-Sample Z**** **

Test of μ = 16.43 vs< 16.43

The assumed standard deviation = 0.8

N Mean SE Mean 95% Upper Bound Z P

15 16.000 0.207 16.340 -2.08 0.019

**conclusion: Rejected at alpha=0.05**

**2**

.**One-Sample Z: C1**** **

Test of μ = 275 vs> 275

The assumed standard deviation = 55

Variable N Mean StDev SE Mean 95% Lower Bound Z P

C1 30 286.2 55.9 10.0 269.6 1.11 0.133

**Conclusion: Accepted at alpha=.025**

**3.**

**One-Sample T: C2**** **

Test of μ = 65 vs> 65

Variable N Mean StDev SE Mean 95% Lower Bound T P

C2 10 67.00 3.20 1.01 65.15 1.98 0.040

**Conclusion: rejected.**

**4.**Test of p = 0.5 vs p ≠ 0.5

Exact

Sample X N Sample p 95% CI P-Value

1 53 100 0.530000 (0.427581, 0.630595) 0.617

**Conclusion:Accepted**

**5.**Test of p = 0.3 vs p ≠ 0.3

Exact

Sample X N Sample p 95% CI P-Value

1 43 150 0.286667 (0.215861, 0.366109) 0.724

**Conclusion: Accepted**