Chapter 8 [8.1, 8.2]

Chapter 8

[8.1, 8.2]

STAT 109 MSU SPRING 2022

8.1 The Essential Ingredients of Hypothesis Testing

​You informally perform hypothesis tests all the time. Hopefully, it's the basis for how you make decisions. Let's try it now.

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**A class activity is not included in these notes. Refer to your own notes or ask a classmate for them.

8.1 The Essential Ingredients of Hypothesis Testing

The Process​

Basically, there are four steps to a hypothesis test. Formally, the process can look a little complicated, but if you remember the basic process, then you'll know how to proceed at each step.

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1. Figure out what you're testing. What claim do you want to support or refute?

2. Gather real data and perform calculations.

3. Determine whether the real data is probable or improbable based on the claim.

4. Draw a conclusion.

8.1 The Essential Ingredients of Hypothesis Testing

1. Hypotheses

Before you begin a hypothesis test, we have to know what claim is being tested, and what it is being tested against. In statistics, we are always testing claims related to the value of a population parameter. For proportions, we are making a claim about a population proportion. There are two hypotheses for our test:

1. The Null Hypothesis: The null hypothesis claims that the population proportion p is equal to some neutral value, p₀. In symbols, H₀: p = p₀

2. The Alternate Hypothesis: The alternate hypothesis presents a claim against the null hypothesis that we would like to demonstrate. We can say that the population p proportion is not equal to ​the neutral assumption p₀ or that p is greater than or less than p₀. In symbols, H_A: p < p₀ or H_A: p ≠​ p₀ or H_A: p > p₀

8.1 The Essential Ingredients of Hypothesis Testing

2. Collect and Standardize a Sample

We test our null hypothesis against a sample we collect under the assumption that the null is true. Take a sample and measure the relevant statistic. Then standardize this data to obtain a test statistic.

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Since we're working with proportions, we collect a sample to get a sample proportion, then find its z-score, which we use as the test statistic.

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Sample --> Sample Proportion (p̂) --> Z-Test Statistic (z-stat)

8.1 The Essential Ingredients of Hypothesis Testing

3. Measure Probability

Remember that we perform a hypothesis test under the assumption that the null hypothesis is true. We use what we know about the Normal distribution to determine whether the sample we gathered is likely or unlikely to have been taken from a population for which the null is true through probability. We call this the p-value.

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​The P-Value of a hypothesis test is the probability of obtaining a test statistic as extreme as or more extreme than the calculated test statistic.

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We use the normal tables to calculate the p-value, and the direction (left or right) depends on the alternate hypothesis.​

8.1 The Essential Ingredients of Hypothesis Testing

More about P-value.​

Two-Tailed Tests: If the alternate hypothesis is H_A: p ≠ p_0, then it is a "two-tailed test." This means that the p-value is measured in both tails of the normal distribution. We measure the area in one tail of the distribution and double it.

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​Left-Tailed Tests: If the alternate hypothesis is H_A: p < p₀, then it is a "left-tailed test." This means that the p-value is measured in the left tail of the normal distribution. Typically, the test statistic is negative in this case.

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Right-Tailed Tests:​ If the alternate hypothesis is H_A: p > p₀, then it is a "right-tailed test." This means that the p-value is measured in the right tail of the normal distribution. Typically, the test statistic is positive in this case.

8.1 The Essential Ingredients of Hypothesis Testing

4. Results of the Test

The p-value of the test tells you how likely it is to obtain the sample used in the test from a population in which the null hypothesis is true. If that probability is extremely low, then it is unlikely that the sample came from the population as expected. Since the sample actually happened, this means we may be wrong about the population parameter we are studying.

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We use a significance level α​ to determine if the p-value is low enough to draw this conclusion.

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If the p-value of the test ​is less than or equal to the significance level, we reject the null hypothesis. We conclude that the probability that the sample came from the assumed population is so low, the population is likely not what we expect.

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​If the p-value of the test is greater than the significance level, we fail to reject the null hypothesis. We conclude that the probability that the sample came from the assumed population is not low enough to reject the null hypothesis. We are not saying that the null hypothesis is true, but we are unable to refute its claim.

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8.1 The Essential Ingredients of Hypothesis Testing

8.2 Hypothesis Testing in Four Steps

8.2 Hypothesis Testing in Four Steps

Multiple Choice

According to the CDC, CoVid is undetectable in 90% of those infected after a period of ten days. A research lab is studying the effects of IV Vitamin C therapy for patients with Covid, which should improve this rate. In their study of 200 cases treated with IVVC, CoVid was undetectable in 94% of cases.

Run a hypothesis test with 5% significance to determine whether the treatment should be considered effective.

STEP 1: Hypotheses. Choose the correct null and alternate hypotheses.

• $$H_0:\ p=0.90$$  

  $$H_A:\ p\ne0.90$$  

• $$H_0:\ p=0.90$$  

  $$H_A:\ p<0.90$$  

• $$H_0:\ p=0.90$$  

  $$H_A:\ p>0.90$$  

Multiple Choice

According to the CDC, CoVid is undetectable in 90% of those infected after a period of ten days. A research lab is studying the effects of IV Vitamin C therapy for patients with Covid, which should improve this rate. In their study of 200 cases treated with IVVC, CoVid was undetectable in 94% of cases.

Run a hypothesis test with 5% significance to determine whether the treatment should be considered effective.

STEP 4: Conclusion. Choose the correct conclusion.

• Reject the null hypothesis because the p-value is less than the significance level.

• Fail to reject the null hypothesis because the p-value is greater than the significance level.