Chapter 9: Hypothesis Testing

Relationship between Interval estimation and Hypothesis Testing

Select a simple random sample from the population and use the value of the sample mean x ̅ to develop the confidence interval for the population mean m. (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized value m0, do not reject H0. Otherwise, reject H0. (Actually, H0 should be rejected if m0 happens to be equal to one of the end points of the confidence interval.) The 97% confidence interval for m is x ̅±z_(α/2) σ/√n=

A Summary of Forms for Null and Alternative Hypotheses About a Population Proportion

The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p0 is the hypothesized value of the population proportion). One-tailed (lower tail) - H_0:p ≥ p_0 H_a:p < p_0 One-tailed (upper tail) - H_0:p ≤ p_0 〖 H〗_a:p > p_0 Two-tailed - H_0:p = p_0 H_a:p ≠ p_0

Type 2 Error

• A Type II error is accepting H0 when it is false. • It is difficult to control the probability of making a Type II error. • Statisticians avoid the risk of making a Type II error by using "do not reject H0" and not "accept H0".

Suggested Guidelines for Interpreting p-Values

• Less than .01 o Overwhelming evidence to conclude Ha is true. • Between .01 and .05 o Strong evidence to conclude Ha is true. • Between .05 and .10 o Weak evidence to conclude Ha is true. • Greater than .10 o Insufficient evidence to conclude Ha is true.

Critical Value Approach to One-Tailed Hypothesis Testing

• The test statistic z has a standard normal probability distribution. • We can use the standard normal probability distribution table to find the z-value with an area of a in the lower (or upper) tail of the distribution. • The value of the test statistic that establishes the boundary of the rejection region is called the critical value for the test. • The rejection rule is: o Lower tail: Reject H0 if z < -za o Upper tail: Reject H0 if z > za

Tests About a Population Mean: s Unknown

Test Statistic: t=(x ̅-μ_0)/(s⁄√n) This test statistic has a t distribution with n - 1 degrees of freedom. Rejection Rule: p -Value Approach Reject H0 if p -value < a Rejection Rule: Critical Value Approach H0: m > m0 Reject H0 if t < -ta H0: m < m0 Reject H0 if t > ta H0: m = m0 Reject H0 if t < - ta/2 or t > ta/2

Tests About a Population Proportion

Test Statistic: z=(p ̅-p_0)/σ_p ̅ where: σ_p ̅ =√((p_0 (1-p_0 ))/n) assuming np > 5 and n(1 - p) > 5 Rejection Rule: p -Value Approach Reject H0 if p -value < a Rejection Rule: Critical Value Approach H0: p < p0 Reject H0 if z > za H0: p > p0 Reject H0 if z < -za H0: p = p0 Reject H0 if z < -za/2 or z > za/2

Summary of Froms for Null Alternative Hypotheses about a Population Mean

The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population mean m must take one of the following three forms (where m0 is the hypothesized value of the population mean). H_0: μ ≥ μ_0 H_0: μ ≤ μ_0 H_0: μ = μ_0 H_a: μ < μ_0 H_a: μ > μ_0 H_a: μ ≠ μ_0 One-tailed (lower-tail) One-tailed (upper-tail) Two-tailed test

Type 1 Error

• Because hypothesis tests are based on sample data, we must allow for the possibility of errors. • A Type I error is rejecting H0 when it is true. • The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. • Applications of hypothesis testing that only control the Type I error are often called significance tests.

p-Value Approach of Two-Tailed Hypothesis Testing

• Compute the p-value using the following three steps: • 1. Compute the value of the test statistic z. • 2. If z is in the upper tail (z > 0), compute the probability that z is greater than or equal to the value of the test statistic. If z is in the lower tail (z < 0), compute the probability that z is less than or equal to the value of the test statistic. • 3. Double the tail area obtained in step 2 to obtain the p-value. o The rejection rule: Reject H0 if the p-value < a .

Hypothesis Testing

• Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. • The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter. • The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis. • The hypothesis testing procedure uses data from a sample to test the two competing statements indicated by H0 and Ha.

Developing Null and Alternative Hypotheses

• It is not always obvious how the null and alternative hypotheses should be formulated. • Care must be taken to structure the hypotheses appropriately so that the test conclusion provides the information the researcher wants. • The context of the situation is very important in determining how the hypotheses should be stated. • In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier. • Correct hypothesis formulation will take practice.

Alternative Hypothesis as a Research Hypothesis

• Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. • In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. • The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected. • Example: o A new teaching method is developed that is believed to be better than the current method. o Alternative Hypothesis: The new teaching method is better. o Null Hypothesis: The new method is no better than the old method. • Example: o A new sales force bonus plan is developed in an attempt to increase sales. o Alternative Hypothesis: The new bonus plan increases sales. o Null Hypothesis: The new bonus plan does not increase sales.

Steps of Hypothesis Testing

• Step 1. Develop the null and alternative hypotheses. • Step 2. Specify the level of significance a. • Step 3. Collect the sample data and compute the value of the test statistic. • p-Value Approach o Step 4. Use the value of the test statistic to compute the p-value. o Step 5. Reject H0 if p-value < a. • Critical Value Approach o Step 4. Use the level of significance a to determine the critical value and the rejection rule. o Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0.

Critical Value Approach to Two-Tailed Hypothesis Testing

• The critical values will occur in both the lower and upper tails of the standard normal curve. • Use the standard normal probability distribution table to find za/2 (the z-value with an area of a/2 in the upper tail of the distribution). • The rejection rule is: Reject H0 if z < -za/2 or z > za/2.

p-Values and the t Distribution

• The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test. • However, we can still use the t distribution table to identify a range for the p-value. • An advantage of computer software packages is that the computer output will provide the p-value for the t distribution.

p-Value Approach to One-Tailed Hypothesis Testing

• The p-value is the probability computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis. • If the p-value is less than or equal to the level of significance a, the value of the test statistic is in the rejection region. • Reject H0 if the p-value < a .

Developing Null and Alternative Hypotheses

• We might begin with a belief or assumption that a statement about the value of a population parameter is true. • We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. • In these situations, it is helpful to develop the null hypothesis first. • Null Hypothesis as an Assumption to be challenged o Ex: The label on a soft drink bottle states that it contains 67.6 fluid ounces. Null Hypothesis: • The label is correct. m > 67.6 ounces. Alternative Hypothesis: • The label is incorrect. m < 67.6 ounces.