MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Hypothesis Testing in Statistics #### Problem Statement Suppose a poll is taken that shows that 510 out of 1000 randomly selected, independent people believe the rich should pay more taxes than they do. Test the hypothesis that a majority (more than 50%) believe the rich should pay more taxes than they do. Use a significance level of 0.05. #### Step 1: State the Null and Alternative Hypotheses Select the appropriate null and alternative hypotheses: - \(H_0\): \(p \leq 0.50\) - \(H_a\): \(p > 0.50\) This represents the hypothesis that the percentage of people who believe the rich should pay more in taxes is more than 50%. #### Step 2: Calculate the Test Statistic Find the value of the test statistic \(z\). \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where: - \(\hat{p}\) is the sample proportion - \(p_0\) is the hypothesized population proportion - \(n\) is the sample size Substitute the given values into the formula and round to two decimal places. #### Step 3: Find the p-value Locate the corresponding p-value for the test statistic \(z\). This will involve looking up the value on a standard normal distribution table or using statistical software. \[ \text{p-value} = P(Z > z) \] Round the p-value to three decimal places as needed. #### Step 4: Make a Decision Compare the p-value to the significance level (0.05): - If the p-value is less than 0.05, reject the null hypothesis. - If the p-value is greater than or equal to 0.05, do not reject the null hypothesis. #### Step 5: Interpret the Results Based on whether or not you reject the null hypothesis, interpret your results: 1. **Reject \(H_0\)**: The percentage of all people who believe the rich should pay more taxes is significantly more than 50%. 2. **Do not reject \(H_0\)**: The percentage of all people who believe the rich should pay more taxes is not significantly more than 50%. #### Graphical Representation If there was a graph or diagram, describe it
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Transcribed Image Text:### Hypothesis Testing in Statistics #### Problem Statement Suppose a poll is taken that shows that 510 out of 1000 randomly selected, independent people believe the rich should pay more taxes than they do. Test the hypothesis that a majority (more than 50%) believe the rich should pay more taxes than they do. Use a significance level of 0.05. #### Step 1: State the Null and Alternative Hypotheses Select the appropriate null and alternative hypotheses: - \(H_0\): \(p \leq 0.50\) - \(H_a\): \(p > 0.50\) This represents the hypothesis that the percentage of people who believe the rich should pay more in taxes is more than 50%. #### Step 2: Calculate the Test Statistic Find the value of the test statistic \(z\). \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where: - \(\hat{p}\) is the sample proportion - \(p_0\) is the hypothesized population proportion - \(n\) is the sample size Substitute the given values into the formula and round to two decimal places. #### Step 3: Find the p-value Locate the corresponding p-value for the test statistic \(z\). This will involve looking up the value on a standard normal distribution table or using statistical software. \[ \text{p-value} = P(Z > z) \] Round the p-value to three decimal places as needed. #### Step 4: Make a Decision Compare the p-value to the significance level (0.05): - If the p-value is less than 0.05, reject the null hypothesis. - If the p-value is greater than or equal to 0.05, do not reject the null hypothesis. #### Step 5: Interpret the Results Based on whether or not you reject the null hypothesis, interpret your results: 1. **Reject \(H_0\)**: The percentage of all people who believe the rich should pay more taxes is significantly more than 50%. 2. **Do not reject \(H_0\)**: The percentage of all people who believe the rich should pay more taxes is not significantly more than 50%. #### Graphical Representation If there was a graph or diagram, describe it
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