Hypothesis Testing for Single Population Proportion and Mean

 

Hypothesis Testing for Single Population Proportion and Mean

Introduction

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves testing an assumption (hypothesis) about a population parameter and deciding whether to reject this assumption based on the sample data. In this note, we will discuss hypothesis testing for a single population proportion and mean, with known and unknown variance.

Hypothesis Testing for Single Population Proportion

Steps in Hypothesis Testing for Proportion

1. State the Hypotheses: Define the null hypothesis ( $H_0$ ) and alternative hypothesis ( $H_a$ ).
2. Choose the Significance Level ( $\alpha$ ) : Commonly set at 0.05.
3. Select the Appropriate Test: Use the z-test for proportion.
4. Compute the Test Statistic: Calculate the z-statistic.
5. Determine the P-value or Critical Value: Use the z-distribution.
6. Make a Decision: Compare the p-value with $\alpha$ or use the critical value approach.
7. Draw a Conclusion: Interpret the results in context.

Example: Testing a Population Proportion

Scenario: A survey claims that 60% of people prefer brand A over brand B. A new survey of 150 people finds that 85 prefer brand A. Test if the true proportion is different from 60% at $\alpha = 0.05$ .

Step-by-Step Solution:

1. State the Hypotheses:

   • $H_0$ : $p = 0.60$
   • $H_a$ : $p \neq 0.60$

2. Choose the Significance Level ( $\alpha$ ) : $\alpha = 0.05$

3. Select the Appropriate Test: Proportion z-test

4. Compute the Test Statistic: $\hat{p} = \frac{85}{150} \approx 0.567$ $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} = \frac{0.567 - 0.60}{\sqrt{\frac{0.60 \times 0.40}{150}}} \approx \frac{-0.033}{0.04} = -0.825$

5. Determine the P-value or Critical Value:

   • Use a standard normal distribution table or software to find the p-value for $z = -0.825$ . The p-value is approximately 0.41.

6. Make a Decision:

   • Since the p-value (0.41) is greater than $\alpha$ (0.05), we fail to reject the null hypothesis.

7. Draw a Conclusion:

   • There is not enough evidence to suggest that the true proportion of people who prefer brand A is different from 60%.

Hypothesis Testing for Single Population Mean (Variance Known)

Steps in Hypothesis Testing for Mean with Known Variance

1. State the Hypotheses: Define the null hypothesis ( $H_0$ ) and alternative hypothesis ( $H_a$ ).
2. Choose the Significance Level ( $\alpha$ ) : Commonly set at 0.05.
3. Select the Appropriate Test: Use the z-test for the mean when the population variance is known.
4. Compute the Test Statistic: Calculate the z-statistic.
5. Determine the P-value or Critical Value: Use the z-distribution.
6. Make a Decision: Compare the p-value with $\alpha$ or use the critical value approach.
7. Draw a Conclusion: Interpret the results in context.

Example: Testing a Population Mean (Variance Known)

Scenario: A company claims that the average life of its light bulbs is 1,000 hours. A sample of 30 bulbs has an average life of 980 hours. The population standard deviation is 50 hours. Test if the average life is different from 1,000 hours at $\alpha = 0.05$ .

Step-by-Step Solution:

1. State the Hypotheses:

   • $H_0$ : $\mu = 1000$
   • $H_a$ : $\mu \neq 1000$

2. Choose the Significance Level ( $\alpha$ ) : $\alpha = 0.05$

3. Select the Appropriate Test: One-sample z-test

4. Compute the Test Statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{980 - 1000}{50 / \sqrt{30}} \approx \frac{-20}{9.13} \approx -2.19$

5. Determine the P-value or Critical Value:

   • Use a standard normal distribution table or software to find the p-value for $z = -2.19$ . The p-value is approximately 0.028.

6. Make a Decision:

   • Since the p-value (0.028) is less than $\alpha$ (0.05), we reject the null hypothesis.

7. Draw a Conclusion:

   • There is strong evidence to suggest that the average life of the light bulbs is different from 1,000 hours.

Hypothesis Testing for Single Population Mean (Variance Unknown)

Steps in Hypothesis Testing for Mean with Unknown Variance

1. State the Hypotheses: Define the null hypothesis ( $H_0$ ) and alternative hypothesis ( $H_a$ ).
2. Choose the Significance Level ( $\alpha$ ) : Commonly set at 0.05.
3. Select the Appropriate Test: Use the t-test for the mean when the population variance is unknown.
4. Compute the Test Statistic: Calculate the t-statistic.
5. **Determine the P-