- In a hypothesis test problem, you may see words such as "the level of significance is 1%." The "1%" is the preconceived or preset
*α*. - The statistician setting up the hypothesis test selects the value of
*α*to use**before**collecting the sample data. **If no level of significance is given, a common standard to use is***α*= 0.05.- When you calculate the
*p*-value and draw the picture, the*p*-value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the hypothesis test left, right, or two tailed. - The
**alternative hypothesis**, ${H}_{a}$, tells you if the test is left, right, or two-tailed. It is the**key**to conducting the appropriate test. *H*_{a}**never**has a symbol that contains an equal sign.**Thinking about the meaning of the***p*-value: A data analyst (and anyone else) should have more confidence that he made the correct decision to reject the null hypothesis with a smaller*p*-value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large*p*-value such as 0.4, as opposed to a*p*-value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.

The following examples illustrate a left-, right-, and two-tailed test.

### Example 9.11

*H _{o}*:

*μ*= 5,

*H*:

_{a}*μ*< 5

Test of a single population mean. *H _{a}* tells you the test is left-tailed. The picture of the

*p*-value is as follows:

*H _{0}*:

*μ*= 10,

*H*:

_{a}*μ*< 10

Assume the *p*-value is 0.0935. What type of test is this? Draw the picture of the *p*-value.

### Example 9.12

*H _{0}*:

*p*≤ 0.2

*H*:

_{a}*p*> 0.2

This is a test of a single population proportion. *H _{a}* tells you the test is

**right-tailed**. The picture of the

*p*-value is as follows:

*H _{0}*:

*μ*≤ 1,

*H*:

_{a}*μ*> 1

Assume the *p*-value is 0.1243. What type of test is this? Draw the picture of the *p*-value.

### Example 9.13

*H _{0}*:

*p*= 50

*H*:

_{a}*p*≠ 50

This is a test of a single population mean. *H _{a}* tells you the test is

**two-tailed**. The picture of the

*p*-value is as follows.

*H _{0}*:

*p*= 0.5,

*H*:

_{a}*p*≠ 0.5

Assume the *p*-value is 0.2564. What type of test is this? Draw the picture of the *p*-value.

### Full Hypothesis Test Examples

### Example 9.14

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 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* = 0.05. Assume that the swim times for the 25-yard freestyle are normal.

The mean throwing distance of a football for Marco, a high school freshman quarterback, is 40 yards, with a standard deviation of two yards. The team coach tells Marco to adjust his grip to get more distance. The coach records the distances for 20 throws. For the 20 throws, Marco’s mean distance was 45 yards. The coach thought the different grip helped Marco throw farther than 40 yards. Conduct a hypothesis test using a preset *α* = 0.05. Assume the throw distances for footballs are normal.

First, determine what type of test this is, set up the hypothesis test, find the *p*-value, sketch the graph, and state your conclusion.

#### Using the TI-83, 83+, 84, 84+ Calculator

Press STAT and arrow over to TESTS. Press 1:Z-Test. Arrow over to Stats and press ENTER. Arrow down and enter 40 for *μ*0 (null hypothesis), 2 for *σ*, 45 for the sample mean, and 20 for *n*. Arrow down to *μ*: (alternative hypothesis) and set it either as <, ≠, or >. Press ENTER. Arrow down to Calculate and press ENTER. The calculator not only calculates the *p*-value but it also calculates the test statistic (*z*-score) for the sample mean. Select <, ≠, or > for the alternative hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate). Press ENTER. A shaded graph appears with test statistic and *p*-value. Make sure when you use Draw that no other equations are highlighted in *Y* = and the plots are turned off.

### Historical Note (Example 9.11)

The traditional way to compare the two probabilities, *α* and the *p*-value, is to compare the critical value (*z*-score from *α*) to the test statistic (*z*-score from data). The calculated test statistic for the *p*-value is –2.08. (From the Central Limit Theorem, the test statistic formula is $z=\frac{\overline{x}-{\mu}_{X}}{(\frac{{\sigma}_{X}}{\sqrt{n}})}$.
For this problem, $\overline{x}$ = 16, *μ _{X}* = 16.43 from the null hypothes is,

*σ*= 0.8, and

_{X}*n*= 15.) You can find the critical value for

*α*= 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 (which demonstrates that α >

*p*-value), reject

*H*. Traditionally, the decision to reject or not reject was done in this way. Today, comparing the two probabilities

_{0}*α*and the

*p*-value is very common. For this problem, the

*p*-value, 0.0187 is considerably smaller than

*α*, 0.05. You can be confident about your decision to reject. The graph shows

*α*, the

*p*-value, and the test statistic and the critical value.

### Example 9.15

A college football coach records the mean weight that his players can bench press as **275 pounds**, with a **standard deviation of 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).

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

### Example 9.16

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 assumed to be from a normal distribution.

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, find the *p*-value, state your conclusion, and identify the Type I and Type II errors.

### Example 9.17

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.

A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.

First, determine what type of test this is, set up the hypothesis test, find the *p*-value, sketch the graph, and state your conclusion.

### Example 9.18

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 not 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.

Marketers believe that 92% of adults in the United States own a cell phone. A cell phone manufacturer believes that number is actually lower. 200 American adults are surveyed, of which, 174 report having cell phones. Use a 5% level of significance. State the null and alternative hypothesis, find the *p*-value, state your conclusion, and identify the Type I and Type II errors.

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 *p*. The distribution for the test is normal. The estimated proportion *p*′ is the proportion of fleas killed to the total fleas found on Fido. This is sample information. The problem gives a preconceived *α* = 0.01, for comparison, and a 95% confidence interval computation. The poem is clever and humorous, so please enjoy it!

### Example 9.19

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?

### Example 9.20

The National Institute of Standards and Technology provides exact data on conductivity properties of materials. Following are conductivity measurements for 11 randomly selected pieces of a particular type of glass.

1.11; 1.07; 1.11; 1.07; 1.12; 1.08; .98; .98 1.02; .95; .95

Is there convincing evidence that the average conductivity of this type of glass is greater than one? Use a significance level of 0.05. Assume the population is normal.

### Example 9.21

In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users (the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type I error.

### Example 9.22

According to the US Census there are approximately 268,608,618 residents aged 12 and older. Statistics from the Rape, Abuse, and Incest National Network indicate that, on average, 207,754 rapes occur each year (male and female) for persons aged 12 and older. This translates into a percentage of sexual assaults of 0.078%. In Daviess County, KY, there were reported 11 rapes for a population of 37,937. Conduct an appropriate hypothesis test to determine if there is a statistically significant difference between the local sexual assault percentage and the national sexual assault percentage. Use a significance level of 0.01.