Suppose that in the past, 94% of all Hispanic grocery shoppers were women. Perhaps due to changing cultural values, we believe that more Hispanic men are now grocery shopping. We randomly sample 689 Hispanic grocery shoppers from around the United States and 606 are women. Does this result provide enough evidence to conclude that a lower proportion of Hispanic grocery shoppers now are women?
606/689 = 88%
That does not help at all.
Do you think a drop of 6% is significant?
To determine if there is enough evidence to conclude that a lower proportion of Hispanic grocery shoppers are women, we can conduct a hypothesis test. Here's how we can approach this problem:
Step 1: State the null hypothesis and alternative hypothesis.
- Null hypothesis (H0): The proportion of Hispanic grocery shoppers who are women is still 94%.
- Alternative hypothesis (Ha): The proportion of Hispanic grocery shoppers who are women is lower than 94%.
Step 2: Define the significance level.
- Let's assume a significance level of 0.05, which means we're willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true).
Step 3: Calculate the test statistic and p-value.
- We can use the z-test for proportions to calculate the test statistic and find the corresponding p-value. The test statistic is given by:
z = (p - P) / sqrt(P * (1 - P) / n)
where p is the sample proportion, P is the hypothesized proportion under the null hypothesis, and n is the sample size.
- In this case, p = 606/689 = 0.88, P = 0.94, and n = 689.
z = (0.88 - 0.94) / sqrt(0.94 * (1 - 0.94) / 689)
Step 4: Determine the critical value and compare with the test statistic.
- Since we are testing whether the proportion of women is lower, we will perform a one-tailed test.
- The critical value can be found using a standard normal distribution table or a calculator.
- At a significance level of 0.05, the critical value corresponds to a z-score of approximately -1.645.
Step 5: Make a decision and interpret the results.
- If the test statistic is less than the critical value, we reject the null hypothesis and conclude that there is enough evidence to suggest a lower proportion of Hispanic grocery shoppers are women.
- If the test statistic is greater than the critical value, we fail to reject the null hypothesis and do not have enough evidence to suggest a lower proportion of Hispanic grocery shoppers are women.
By following these steps, you can determine whether the result provides enough evidence to conclude that a lower proportion of Hispanic grocery shoppers are women.