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Chapter 10

One-Sample Tests of Hypothesis

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In this chapter, we develop a procedure to test the validity of a statement about a population parameter. We begin by defining a hypothesis and hypothesis testing. Next, the steps in hypothesis testing are outlined. The we conduct tests of hypothesis testing for means. Finally, we describe possible errors due to sampling in hypothesis testing.

Learning Objectives

10-1 Explain the process of testing a hypothesis

10-2 Apply the six-step procedure for testing a hypothesis

10-3 Distinguish between a one-tailed and a two- tailed test of hypothesis

10-4 Conduct a test of a hypothesis about a population mean

10-5 Compute and interpret a p-value

10-6 Use a t-statistic to test a hypothesis

10-7 Compute the probability of a Type II error

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Hypothesis Testing (1 of 3)

Hypothesis testing begins with a hypothesis statement about a population parameter

HYPOTHESIS A statement about a population parameter subject to verification

Examples

The mean speed of automobiles passing milepost 150 on the West Virginia Turnpike is 68 mph

The mean cost to remodel a kitchen is $20,000

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Here are examples of statements we might want to test.

Hypothesis Testing (2 of 3)

The objective of hypothesis testing is to verify the validity of a statement about a population parameter

HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.

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There are six steps in the hypothesis testing procedure. We will discuss each of the steps in detail.

Hypothesis Testing (3 of 3)

Step 1: State null and alternate hypotheses

Step 2: Select a level of significance

Step 3: Identify the test statistic

Step 4: Formulate a decision rule

Step 5: Take a sample, arrive at decision

Step 6: Interpret the result

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There are six steps in the hypothesis testing procedure. We will discuss each of the steps in detail.

Step 1 of the Six-Step Process (1 of 2)

State the null hypothesis (H0) and the alternate hypothesis (H1)

NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence.

The null hypothesis always includes the equal sign

For example; =, ≥, or ≤ will be used in H0

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The null hypothesis is a statement that is not rejected unless our sample data provide convincing evidence that it is false. Failing to reject the null hypothesis does not prove that H0 is true; it means we have failed to disprove H0. H1 is what you will conclude if you reject the null hypothesis. We turn to the alternate hypothesis only if the data suggest the null hypothesis is untrue.

Step 1 of the Six-Step Process (2 of 2)

ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.

The alternate hypothesis never includes the equal sign

For example; ≠, <, or > is used in H1

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Step 2 of the Process

Next, you select the level of significance, α

LEVEL OF SIGNIFICANCE The probability of rejecting the null hypothesis when it is true.

Sometimes called the level of risk

Can be any value between 0 and 1

Traditionally,

.05 is used for consumer research projects

.01 for quality assurance

.10 for political polling

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Possible Error in Hypothesis Testing (1 of 2)

Since the researcher cannot study every item or individual in the population, error is possible

TYPE I ERROR Rejecting the null hypothesis, H0,when it is true.

Type I error is designated with the Greek letter alpha, α

TYPE II ERROR Not rejecting the null hypothesis when it is false.

Type II error is designated with the Greek letter beta, β

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Possible Error in Hypothesis Testing (2 of 2)

Null Hypothesis	Researcher: Does Not Reject H0	Researcher: Rejects H0
H0 is true	Correct decision	Type l error
H0 is false	Type ll error	Correct decision

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Step 3 of the Process (1 of 2)

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There are many test statistics. In this chapter we use z and t as the test statistic, based on whether or not we know the population standard deviation. Use formula (10-1) when you know the population standard deviation.

Step 3 of the Process (2 of 2)

We can determine whether the distance between

is statistically significant by finding the number of standard deviations

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Step 4 of the Process

Formulate the decision rule

The decision rule is a statement of specific conditions under which the null hypothesis is rejected and the conditions under which it is not rejected

The region or area of rejection defines the location of all the values that are either so large or so small that their probability of occurrence under a true null hypothesis is remote

CRITICAL VALUE The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

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See the next slide for an explanation of the critical value.

Critical Value (1 of 2)

The sampling distribution of the statistic z follows the normal distribution

Here, an α of .05 is used in a one-tailed test

The value 1.645 separates the regions where the null hypothesis is rejected and where it is not rejected

The value 1.645 is the critical value

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A one-tailed test (right-tailed) is used in this chart; this and how we obtain the critical value will be covered in more detail later. The area where the null hypothesis is not rejected is to the left of 1.645 and the area of rejection is to the right of 1.645.

Critical Value (2 of 2)

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Steps 5 & 6 of the Six-Step Process

Step 5 Make a decision

First, select a sample and compute the value of the test statistic

Compare the value of the test statistic to the critical value

Then, make the decision regarding the null hypothesis

Step 6 Interpret the results

What can we say or report based on the results of the statistical test?

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One-Tailed and Two-Tailed Tests (1 of 3)

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One-Tailed and Two-Tailed Tests (2 of 3)

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One-Tailed and Two-Tailed Tests (3 of 3)

Note that the total area in the normal distribution is 1.0000.

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Two-Tailed Test Example, σ Known (1 of 6)

Jamestown Steel Company manufactures and assembles desks and other office equipment at several plants in New York State. At the Fredonia plant, the weekly production of the Model A325 desk follows a normal distribution with a mean of 200 and a standard deviation of 16. New production methods have been introduced and the vice president of manufacturing would like to investigate whether there has been a change in weekly production of the Model A325. Is the mean number of desks produced different from 200 at the .01 significance level?

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This is a two-tailed test because the alternate hypothesis does not state a direction. The level of significance, .01, is given. When testing a hypothesis about a population mean, if the population follows the normal distribution and the population standard deviation is known, the test statistic is z and is determined from formula 10-1.

Two-Tailed Test Example, σ Known (2 of 6)

Step 1: State the null hypothesis and alternate hypothesis.

Step 2: Select the level of significance. Here α = .01

Step 3: Select the test statistic. In this example, we’ll use z

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Two-Tailed Test Example, σ Known (3 of 6)

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Two-Tailed Test Example, σ Known (4 of 6)

Step 4: Formulate the decision rule by first determining the critical values of z.

Decision Rule: If the computed value of z is not between -2.575 and 2.576, reject the null hypothesis. If z falls between -2.576 and 2.576, do not reject the null hypothesis.

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Two-Tailed Test Example, σ Known (5 of 6)

Step 5: Take sample, compute the test statistic, make decision.

The mean number of desks produced last year (50 weeks because the plant was shut down 2 weeks for vacation) is 203.5. The standard deviation of the population is 16 desks per week. Compute z with formula 10-1.

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Tell the vice president the sample information fails to indicate that the new production methods resulted in a change in the 200-desks-per-week production rate. However, we did not prove the assembly rate is still 200 per week; we failed to disprove it, which is not the same thing as proving it to be true.

Two-Tailed Test Example, σ Known (6 of 6)

Decision: Because 1.547 does not fall in the rejection region, we decide not to reject H0.

Step 6: Interpret the result.

We did not reject the null hypothesis, so we have failed to show that the population mean has changed from 200 per week.

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One-Tailed Test (1 of 2)

Suppose instead of wanting to know if there had been a change in the mean number of desks assembled, the vice president wanted to know if there had been an increase in the number of units assembled. Can we conclude, because of the improved production methods, that the mean number of desks assembled in the last 50 weeks was more than 200? Use

α= .01

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One-Tailed Test (2 of 2)

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The p-Value in Hypothesis Testing (1 of 2)

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A p-value is the probability that the value of the test statistic is as extreme as the value computed, when the null hypothesis is true. A very small p-value, such as .001, indicates that there is little likelihood that H0 is true. On the other hand, a p-value of .2033 means that H0 is not rejected, and there is little likelihood that it is false.

The p-Value in Hypothesis Testing (2 of 2)

INTERPRETING THE WEIGHT OF EVIDENCE AGAINST H0

If the p-value is less than

.10, we have some evidence that H0 is not true.

.05, we have strong evidence that H0 is not true.

.01, we have very strong evidence that H0 is not true.

.001, we have extremely strong evidence that H0 is not true.

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Finding a p-Value (1 of 2)

In the previous example about desk production, the computed z was 1.547 and H0 was not rejected

Round the computed z-value to two decimal places, 1.55

Using the z-table, find the probability of finding a z-value of 1.55 or more by .5000 - .4394 = .0606

Since this is a two-tailed test 2(.0606) = .1212

In this chart, we can easily compare the p-value with the level of significance

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A p-value is the probability that the value of the test statistic is as extreme as the value computed, when the null hypothesis is true. The probability of finding a sample mean greater than 203.5 when the population mean is 200 is .0606. The two-tailed p-value is 2(.0606) = .1212. Thus, the p-value of .1212 is greater than the significance level of .01 so H0 is not rejected.

Finding a p-Value (2 of 2)

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Hypothesis Testing, σ Unknown

When testing a hypothesis about a population mean

The major characteristics of the t distribution are

It is a continuous distribution

It is bell-shaped and symmetrical

There is a family of t distributions, depending on the number of degrees of freedom

It is flatter, or more spread out, than the standard normal distribution

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Hypothesis Testing, σ Unknown Example (1 of 6)

The McFarland Insurance Company Claims Department reports the mean cost to process a claim is $60. An industry comparison showed this amount to be larger than most other insurance companies, so the company instituted cost-cutting measures. To evaluate the cost-cutting measures, a random sample was taken of 26 claims processed last month and the cost to process each claim was recorded (see below).

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We use the six-step hypothesis testing procedure. This is a one-tailed test, a left-tailed test.

Hypothesis Testing, σ Unknown Example (2 of 6)

At the .01 significance level, is it reasonable to conclude the mean cost to process a claim is now less than $60?

Step 1: State the null hypothesis and the alternate hypothesis

$45	$49	$62	$40	$43	$61
48	53	67	63	78	64
48	54	51	56	63	69
58	51	58	59	56	57
38	76

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Hypothesis Testing, σ Unknown Example (3 of 6)

Step 2: Select the level of significance; we will use .01

Step 3: Select the test statistic; we will use t

Step 4: Formulate the decision rule; reject H0 if t is less than -2.485

Step 5: Take sample, make decision; Do not reject H0

Step 6: Interpret the result; The test results do not allow the claims manager to conclude the cost-cutting measures have been effective.

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Hypothesis Testing, σ Unknown Example (4 of 6)

		Confidence Intervals
	80%	90%	95%	98%	99%	99.9%
		Level of Significance for One-Tailed Test, α
df	0.10	0.05	0.025	0.010	0.005	0.0005
		Level of Significance for Two-Tailed Test, α
	0.20	0.10	0.05	0.02	0.01	0.0001
.	.	.	.	.	.	.
.	.	.	.	.	.	.
.	.	.	.	.	.	.
21	1.323	1.721	2.080	2.518	2.831	3.819
22	1.321	1.717	2.074	2.508	2.819	3.792
23	1.319	1.714	2.069	2.500	2.807	3.768
24	1.318	1.711	2.064	2.492	2.797	3.745
25	1.316	1.708	2.060	2.485	2.787	3.725

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Hypothesis Testing, σ Unknown Example (5 of 6)

		Confidence Intervals
	80%	90%	95%	98%	99%	99.9%
		Level of Significance for One-Tailed Test, α
df	0.10	0.05	0.025	0.010	0.005	0.0005
		Level of Significance for Two-Tailed Test, α
	0.20	0.10	0.05	0.02	0.01	0.0001
.	.	.	.	.	.	.
.	.	.	.	.	.	.
.	.	.	.	.	.	.
26	1.315	1.706	2.056	2.479	2.779	3.707
27	1.314	1.703	2.052	2.473	2.771	3.690
28	1.313	1.701	2.048	2.467	2.763	3.674
29	1.311	1.699	2.045	2.462	2.756	3.659
30	1.310	1.697	2.042	2.457	2.750	3.646

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Hypothesis Testing, σ Unknown Example (6 of 6)

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Type I and Type II Errors (1 of 2)

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Type I and Type II Errors (2 of 2)

The likelihood of a Type II error must be calculated comparing the hypothesized distribution to an alternate distribution based on sample results and can be calculated with this formula

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Type II Error Example (1 of 3)

Western Wire Products purchases steel bars to make cotter pins. Past experience indicates that the mean tensile strength of all incoming shipments is 10,000 psi and that the standard deviation is 400 psi. To monitor the quality of the cotter pins, samples of 100 pins are randomly selected and tested for their strength. Using a 0.05 significance level, accept the shipment if the sample mean strength falls between the critical values 9.922 psi and 10.078 psi. If the sample mean does not fall between the critical values, we conclude the shipment does not meet the quality standard.

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Type II Error Example (2 of 3)

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Type II Error Example (3 of 3)

The sample mean, 9.900 psi, is not within the specified range. To calculate the probability of a Type II error, assume the sample mean is the true mean (see graph B). Determine the probability of the sample mean falling between 9.900 and 9.922. Then subtract this probability from .5000 to arrive at the probability of making a Type II error, .2912

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End of Presentation

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