Chapter Seven Hypothesis Testing with ONE Sample

Section 7.1 Introduction to Hypothesis Testing

Hypothesis Tests … A process that uses sample statistics to test a claim about a population parameter. Test includes: ◦ Stating a NULL and an ALTERNATIVE Hypothesis. ◦ Determining whether to REJECT or to NOT REJECT the Null Hypothesis. (If the Null is rejected, that means the Alternative must be true.)

Stating a Hypothesis The Null Hypothesis (H 0 ) is a statistical hypothesis that contains some statement of equality, such as =, The Alternative Hypothesis (H a ) is the complement of the null hypothesis. It contains a statement of inequality, such as ≠,

Left, Right, or Two-Tailed Tests If the Alternative Hypotheses, H a, includes <, it is considered a LEFT TAILED test. If the Alternative Hypotheses, H a, includes >, it is considered a RIGHT TAILED test. If the Alternative Hypotheses, H a, includes ≠, it is considered a TWO TAILED test.

EX: State the Null and Alternative Hypotheses. *As stated by a company’s shipping department, the number of shipping errors per mission shipments has a standard deviation that is less than 3. *A state park claims that the mean height of oak trees in the park is at least 85 feet.

Types of Errors When doing a test, you will decide whether to reject or not reject the null hypothesis. Since the decision is based on SAMPLE data, there is a possibility the decision will be wrong. Type I error: the null hypothesis is rejected when it is true. Type II error: the null hypothesis is not rejected when it is false.

4 possible outcomes… TRUTH OF H 0 DECISIONH 0 is TRUEH 0 is FALSE Do not reject H 0 Correct Decision Type II Error Reject H 0 Type I ErrorCorrect Decision

Level of Significance The level of significance is the maximum allowed probability of making a Type I error. It is denoted by the lowercase Greek letter alpha ( α). The probability of making a Type II error is denoted by the lowercase Greek letter beta ( β).

Section 7.2 Hypothesis Testing for the MEAN (Large Samples)

p-Values If the null hypothesis is true, a p- Value of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. The p-Value is connected to the area under the curve to the left and/or right on the normal curve.

Finding the p-Value for a Hypothesis Test – using the table To find p-Value ◦ Left tailed: p = area in the left tail ◦ Right tailed: p = area in the right tail ◦ Two Tailed: p = 2(area in one of the tails) This section we’ll be finding the z-values and using the standard normal table.

Using p-Values for a z-Test Z-Test used when the population is normal, σ is known, and n is at least 30. If n is more than 30, we can use s for σ.

Making and Interpreting your Decision Decision Rule based on the p-Value Compare the p-Value with alpha. ◦ If p < α, reject H 0 ◦ If p > α, do not reject H 0

Find the p-value. Decide whether to reject or not reject the null hypothesis Left tailed test, z = -1.55, α = 0.05 Two tailed test, z = 1.23, α = 0.10

General Steps for Hypothesis Testing – P Value METHOD 1. State the null and alternative hypotheses. 2. Specify the level of significance, α 3. Sketch the curve. 4. Find the standardized statistic add to sketch and shade. (usually z or t-score) 5. Find the p-Value 6. Compare p-Value to alpha to make the decision. 7. Write a statement to interpret the decision in context of the original claim.

A manufacturer of sprinkler systems designed for fire protection claims the average activating temperature is at least 135 o F. To test this claim, you randomly select a sample of 32 systems and find mean = 133, and s = 3.3. At α = 0.10, do you have enough evidence to reject the manufacturer’s claim?

Rejection Regions & Critical Values The Critical value (z 0 ) is the z-score that corresponds to the level of significance (alpha) Z 0 separates the rejection region from the non-rejection region Sketch a normal curve and shade the rejection region. (Left, right, or two tailed)

Find z 0 and shade rejection region Right tailed test, alpha = 0.08 Two tailed test, alpha = 0.10

Guidelines – using rejection regions 1. find H 0 and H a 2. identify alpha 3. find z 0 – the critical value(s) 4. shade the rejection region(s) 5. find z 6. make decision (Is z in the rejection region?) 7. interpret decision

A fast food restaurant estimates that the mean sodium content in one of its breakfast sandwiches is no more than 920 milligrams. A random sample of 44 sandwiches has a mean sodium content of 925 with s = 18. At alpha = 0.10, do you have enough evidence to reject the restaurant’s claim?