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**Hypothesis Testing with ONE Sample**

Chapter Seven Hypothesis Testing with ONE Sample

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**Introduction to Hypothesis Testing**

Section 7.1 Introduction to Hypothesis Testing

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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.)

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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 >

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**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.

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**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.

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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.

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**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

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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.

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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.

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**Making and Interpreting your Decision**

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

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**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.

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**Hypothesis Testing for the MEAN (Large Samples)**

Section 7.2 Hypothesis Testing for the MEAN (Large Samples)

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**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

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**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.

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**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

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**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.

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**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

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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?

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**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)

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**Find z0 and shade rejection region**

18. Right tailed test, alpha = 0.08 22. Two tailed test, alpha = 0.10

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**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

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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?

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