 # Hypothesis Testing with ONE Sample

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Hypothesis Testing with ONE Sample
Chapter Seven Hypothesis Testing with ONE Sample

Introduction to Hypothesis Testing
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 (H0) is a statistical hypothesis that contains some statement of equality, such as =, <, or > The Alternative Hypothesis (Ha) is the complement of the null hypothesis. It contains a statement of inequality, such as ≠, <, or >

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

EX: State the Null and Alternative Hypotheses.
26. As stated by a company’s shipping department, the number of shipping errors per mission shipments has a standard deviation that is less than 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 H0 DECISION H0 is TRUE H0 is FALSE
Do not reject H0 Correct Decision Type II Error Reject H0 Type I Error

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. The probability of making a Type II error is denoted by the lowercase Greek letter beta.

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.

Decision Rule based on the p-Value Compare the p-Value with alpha. If p < alpha, reject H0 If p > alpha, do not reject H0

General Steps for Hypothesis Testing
State the null and alternative hypotheses. Specify the level of significance. Sketch the curve. Find the standardized statistic add to sketch and shade. (usually z or t-score) Find the p-Value Compare p-Value to alpha to make the decision. Write a statement to interpret the decision in context of the original claim.

Hypothesis Testing for the MEAN (Large Samples)
Section 7.2 Hypothesis Testing for the MEAN (Large Samples)

Using p-Value to Make Decisions
Decision Rule based on the p-Value Compare the p-Value with alpha. If p < alpha, reject H0 If p > alpha, do not reject H0

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.

Find the p-value. Decide whether to reject or not reject the null hypothesis
4. Left tailed test, z = -1.55, alpha = 0.05 8. Two tailed test, z = 1.23, alpha = 0.10

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

Guidelines – using the p-value
1. find H0 and Ha 2. identify alpha 3. find z 4. find area that corresponds to z (the p-value) 5. compare p-value to alpha 6. make decision 7. interpret decision

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

Rejection Regions & Critical Values
The Critical value (z0) is the z-score that corresponds to the level of significance (alpha) Z0 separates the rejection region from the non-rejection region Sketch a normal curve and shade the rejection region. (Left, right, or two tailed)

Find z0 and shade rejection region
18. Right tailed test, alpha = 0.08 22. Two tailed test, alpha = 0.10

Guidelines – using rejection regions
1. find H0 and Ha 2. identify alpha 3. find z0 – 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

38. 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?