A telephone company representative estimates that 40% of its customers use voice mail. To test this hypothesis, she selected a sample of 100 customers and found that 37 used voice mail. At a standard deviation of 0.01, is there enough evidence to show the proportion differs from 40%?
Null hypothesis:
p = .40
Alternate hypothesis:
p does not equal .40
You can use a one-sample proportional z-test for your data. (Test sample proportion = .37 and sample size = 100) Find the critical value in the appropriate table at .01 level of significance for a two-tailed test. Compare the test statistic you calculate to the critical value from the table. If the test statistic exceeds the critical value, reject the null. If the test statistic does not exceed the critical value, do not reject the null. You can draw your conclusions from there.
To determine if there is enough evidence to show that the proportion differs from 40%, we can use a hypothesis test. Specifically, we can use a one-sample proportion test.
The hypothesis test follows these steps:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis (H0): The proportion of customers using voice mail is equal to 40%.
- Alternative hypothesis (Ha): The proportion of customers using voice mail differs from 40%.
Step 2: Select a significance level (α):
- The significance level (α) determines how confident we want to be in our results. Common values are 0.05 (5%) or 0.01 (1%). For this case, let's assume α = 0.05.
Step 3: Test statistic and distribution:
- We will use the standard normal distribution and the z-test statistic for proportions.
Step 4: Calculate the test statistic:
- The formula to calculate the z-test statistic for proportions is:
z = (p - P) / sqrt(P(1-P)/n)
where:
p = Sample proportion
P = Hypothesized population proportion
n = Sample size
In this case:
- p = 37/100 = 0.37 (sample proportion, customers using voice mail)
- P = 0.40 (hypothesized population proportion)
- n = 100 (sample size)
Step 5: Calculate the critical value and the p-value:
- The critical value(s) separate the rejection region(s) from the non-rejection region(s). We will compare the test statistic to the critical value(s) to make a decision.
- The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.
Step 6: Make a decision and interpret the results:
- If the test statistic falls within the critical region or if the p-value is smaller than the significance level (α), we reject the null hypothesis. This indicates enough evidence to suggest that the proportion differs from 40%.
- If the test statistic falls outside the critical region or if the p-value is greater than the significance level (α), we fail to reject the null hypothesis. This suggests that we do not have enough evidence to conclude that the proportion differs from 40%.
To calculate the test statistic and make a decision:
Step 4: Calculate the test statistic:
z = (p - P) / sqrt(P(1-P)/n)
= (0.37 - 0.40) / sqrt(0.40*(1-0.40)/100)
= -0.03 / sqrt(0.24/100)
= -0.03 / 0.0490
= -0.6122
Step 5: Calculate the critical value and the p-value:
- We will use a two-tailed test since the alternative hypothesis is that the proportion differs from 40%.
- The critical values for a two-tailed test at α = 0.05 are approximately ±1.96.
- The p-value can be calculated using a normal distribution table or statistical software. In this case, the p-value is 2 * (Area to the left of -0.6122).
Step 6: Make a decision and interpret the results:
- The test statistic (-0.6122) falls within the non-rejection region (-1.96 to +1.96), and the p-value is greater than 0.05.
- Therefore, we fail to reject the null hypothesis.
- This means that there is not enough evidence to show that the proportion of customers using voice mail differs from 40% at a significance level of 0.05.
In conclusion, based on the given information, we do not have enough evidence to suggest that the proportion of customers using voice mail differs from 40%.