[1.5%] TWO SAMPLE HYPOTHESIS TEST FOR PROPORTION: Before starting his campaign for mayor, Mr. Emory Board decided to do a study to see if there was a difference in the proportion of registered men and women voters who actually vote (so he would know whom to target his campaign toward). Of the 100 men and 150 women surveyed, 50 men and 100 women admitted to voting. What can he conclude at the .05 level of significance?
4a) Direction = UPPER TAILED / LOWER TAILED / 2-TAILED (circle one)
4b) P-value = ________________
4c) REJECT THE NULL or DO NOT REJECT THE NULL? (circle one)
4d) Can we conclude a difference in voting patterns for men vs. women?
YES or NO (circle one)
To answer this question, we will perform a two-sample hypothesis test for proportions. Here are the steps involved in conducting this test:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0) assumes that there is no difference between the proportion of registered men and women voters who actually vote. The alternative hypothesis (Ha) assumes that there is a difference.
H0: p1 = p2 (The proportion of men who vote is equal to the proportion of women who vote)
Ha: p1 ≠ p2 (The proportion of men who vote is not equal to the proportion of women who vote)
Step 2: Determine the significance level (α):
In this case, the significance level is given as 0.05.
Step 3: Collect the data and calculate the test statistic:
From the given information, we can calculate the sample proportions for men and women who voted:
p̂1 (proportion of men who voted) = 50/100 = 0.5
p̂2 (proportion of women who voted) = 100/150 ≈ 0.667
Next, we calculate the standard error (SE) using the formula:
SE = sqrt( (p̂1 * (1-p̂1) / n1) + (p̂2 * (1-p̂2) / n2) )
where n1 and n2 are the sample sizes (100 and 150 respectively).
Step 4: Calculate the test statistic (z-score):
The test statistic for two-sample hypothesis tests for proportions follows a standard normal distribution. The formula for calculating the test statistic (z-score) is:
z = (p̂1 - p̂2) / SE
Step 5: Determine the critical value(s) or p-value:
Since the alternative hypothesis is two-tailed (p1 ≠ p2), we need to calculate the p-value associated with the test statistic.
For an upper-tailed test (Ha: p1 > p2), the p-value is calculated as P(Z > z).
For a lower-tailed test (Ha: p1 < p2), the p-value is calculated as P(Z < z).
For a two-tailed test (Ha: p1 ≠ p2), the p-value is calculated as 2 * P(Z > |z|).
To find the critical values or p-value, we refer to the standard normal distribution table or use statistical software.
Step 6: Make a conclusion:
Finally, we compare the p-value calculated in Step 5 with the significance level (α). If the p-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now let's apply these steps to the given information:
Given:
n1 = 100 (number of men surveyed)
n2 = 150 (number of women surveyed)
p̂1 = 0.5 (proportion of men who admitted voting)
p̂2 = 0.667 (proportion of women who admitted voting)
Step 1: The null and alternative hypotheses:
H0: p1 = p2
Ha: p1 ≠ p2
Step 2: Chosen significance level:
α = 0.05
Step 3: Calculate the sample proportions and the standard error:
p̂1 = 0.5
p̂2 = 0.667
SE = sqrt((0.5 * (1-0.5) / 100) + (0.667 * (1-0.667) / 150))
Step 4: Calculate the test statistic (z-score):
z = (0.5 - 0.667) / SE
Step 5: Calculate the p-value or find the critical value(s):
Since the alternative hypothesis is two-tailed (Ha: p1 ≠ p2), we need to calculate the p-value as 2 * P(Z > |z|). We can use statistical software or a standard normal distribution table to find the p-value.
Step 6: Make a conclusion:
Compare the calculated p-value with the significance level (α). If the p-value is less than α (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Unfortunately, the actual value of the p-value has not been provided in the question, so I cannot determine whether to reject or fail to reject the null hypothesis. The conclusion will depend on the p-value obtained from the calculations or the critical value(s) compared to the significance level (α).
However, if the calculated p-value is less than 0.05, we can conclude that there is a significant difference in voting patterns between men and women. If the p-value is greater than or equal to 0.05, we cannot conclude a significant difference in voting patterns.
4a) Based on the problem description, it is not specified whether Mr. Emory Board is expecting a specific direction of difference between the proportion of registered men and women voters who actually vote. Therefore, we should assume a 2-tailed test.
4b) To determine the p-value, we need to perform calculations using a two-sample proportion test. Let's calculate the p-value.
First, let's calculate the proportions of men and women who admitted to voting:
Proportion of men who admitted to voting = number of men who admitted to voting / total number of men surveyed = 50 / 100 = 0.5
Proportion of women who admitted to voting = number of women who admitted to voting / total number of women surveyed = 100 / 150 = 0.6667
Now, let's calculate the standard error (SE) for the difference in proportions:
SE = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
= sqrt((0.5 * (1 - 0.5) / 100) + (0.6667 * (1 - 0.6667) / 150))
= sqrt(0.0025/100 + 0.0011111/150)
≈ sqrt(0.000025 + 0.000007407)
Now, let's calculate the test statistic (Z-score):
Z = (p1 - p2) / SE
= (0.5 - 0.6667) / sqrt(0.000032407)
≈ (-0.1667) / 0.005694
≈ -29.234
Since this is a 2-tailed test, we need to find the probability of observing a test statistic as extreme or more extreme than the one calculated. So, we will find the p-value by looking up the Z-score in a standard normal distribution table.
The p-value is extremely small (essentially 0) since the Z-score is very far in the tails of the standard normal distribution table.
4c) Since the p-value is less than the significance level of 0.05, we reject the null hypothesis.
4d) Based on the results, we can conclude that there is a difference in voting patterns for men versus women.