Hypothesis testing is a fundamental aspect of inferential statistics that allows us to make decisions or inferences about population parameters based on sample data. This module will cover the basic concepts, steps involved in hypothesis testing, types of errors, and practical examples to solidify your understanding.

1. Null Hypothesis (H0): A statement that there is no effect or no difference, and it serves as the default or starting assumption.
2. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis, indicating the presence of an effect or a difference.
3. Significance Level (α): The probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05.
4. Test Statistic: A standardized value calculated from sample data, used to decide whether to reject the null hypothesis.
5. P-value: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
6. Critical Value: A threshold value that the test statistic must exceed to reject the null hypothesis.

1. State the Hypotheses:

   • Null Hypothesis (H0): There is no effect or difference.
   • Alternative Hypothesis (H1): There is an effect or difference.

2. Choose the Significance Level (α):

   • Common choices are 0.05, 0.01, or 0.10.

3. Select the Appropriate Test:

   • Depending on the data and the hypothesis, choose a test (e.g., t-test, z-test, chi-square test).

4. Calculate the Test Statistic:

   • Use the sample data to compute the test statistic.

5. Determine the P-value or Critical Value:

   • Compare the test statistic to the critical value or use the p-value to make a decision.

6. Make a Decision:

   • If the test statistic exceeds the critical value or if the p-value is less than α, reject the null hypothesis.
   • Otherwise, do not reject the null hypothesis.

1. Type I Error (α): Rejecting the null hypothesis when it is true.
2. Type II Error (β): Failing to reject the null hypothesis when it is false.

Suppose a company claims that the average weight of their product is 500 grams. A quality control manager wants to test this claim. A sample of 30 products has an average weight of 495 grams with a standard deviation of 10 grams. Test the claim at a 5% significance level.

1. State the Hypotheses:

   • H0: μ = 500 grams
   • H1: μ ≠ 500 grams

2. Choose the Significance Level (α):

   • α = 0.05

3. Select the Appropriate Test:

   • Since the sample size is less than 30 and the population standard deviation is unknown, use a one-sample t-test.

4. Calculate the Test Statistic:

   • \( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \)
   • \( t = \frac{495 - 500}{10 / \sqrt{30}} \)
   • \( t = \frac{-5}{1.825} \)
   • \( t = -2.74 \)

5. Determine the Critical Value:

   • For a two-tailed test with df = 29 and α = 0.05, the critical t-value is approximately ±2.045.

6. Make a Decision:

   • Since -2.74 < -2.045, we reject the null hypothesis.

There is sufficient evidence at the 5% significance level to reject the claim that the average weight of the product is 500 grams.

A researcher claims that the average time spent on a website is 15 minutes. A sample of 50 users has an average time of 14 minutes with a standard deviation of 5 minutes. Test the claim at a 1% significance level.

Solution:

1. H0: μ = 15 minutes
2. H1: μ ≠ 15 minutes
3. α = 0.01
4. Use a one-sample t-test.
5. Calculate the test statistic:
   • \( t = \frac{14 - 15}{5 / \sqrt{50}} \)
   • \( t = \frac{-1}{0.707} \)
   • \( t = -1.414 \)
6. Critical value for df = 49 and α = 0.01 (two-tailed) is approximately ±2.68.
7. Since -1.414 > -2.68, we do not reject the null hypothesis.

A factory claims that the defect rate of their products is 2%. A sample of 200 products shows 10 defects. Test the claim at a 5% significance level.

Solution:

1. H0: p = 0.02
2. H1: p ≠ 0.02
3. α = 0.05
4. Use a one-sample z-test for proportions.
5. Calculate the test statistic:
   • \( \hat{p} = \frac{10}{200} = 0.05 \)
   • \( z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}} \)
   • \( z = \frac{0.05 - 0.02}{\sqrt{\frac{0.02(0.98)}{200}}} \)
   • \( z = \frac{0.03}{0.0098} \)
   • \( z = 3.06 \)
6. Critical value for α = 0.05 (two-tailed) is approximately ±1.96.
7. Since 3.06 > 1.96, we reject the null hypothesis.

• Misinterpreting the p-value: Remember that a p-value less than α indicates strong evidence against the null hypothesis.
• Ignoring assumptions: Ensure the assumptions of the test (e.g., normality, sample size) are met.
• Confusing Type I and Type II errors: Understand the difference and the implications of each type of error.

Hypothesis testing is a powerful tool for making inferences about population parameters based on sample data. By understanding the steps involved, the types of errors, and practicing with real-world examples, you can effectively apply hypothesis testing in various scenarios. In the next module, we will delve into confidence intervals, which provide another method for making inferences about population parameters.