In 2000, National data suggested that 26% of all U.S. businesses were owned by women. In 2009, a consulting firm surveyed a random sample of 700 U.S. businesses and found that 28% of them were owned by women. Is this evidence of an increase in the percentage of businesses that are owned by women in the U.S?

Question

In 2000, National data suggested that 26% of all U.S. businesses were owned by women. In 2009, a consulting firm surveyed a random sample of 700 U.S. businesses and found that 28% of them were owned by women. Is this evidence of an increase in the percentage of businesses that are owned by women in the U.S?

Write your hypothesis on a separate sheet of paper and perform the test.

Find the P-value for the test and, based on your P-value, what should be your decision for H_{0} H0.

Answer 1

To determine if there is evidence of an increase in the percentage of businesses owned by women in the U.S., we can conduct a hypothesis test. The null hypothesis, denoted as H_0, assumes that there is no change in the percentage of businesses owned by women. The alternative hypothesis, denoted as H_a, suggests that there is an increase in the percentage.

H_0: p = 0.26 (proportion of businesses owned by women remains 26%)
H_a: p > 0.26 (proportion of businesses owned by women increases)

To perform the test, we will use a one-sample proportion test. The test statistic for this test is the z-score, which measures how many standard deviations away our observed sample proportion is from the assumed population proportion.

To find the P-value for the test, we need to calculate the z-score and then use a standard normal distribution table or a calculator/statistical software to find the corresponding P-value.

First, calculate the test statistic (z-score) using the formula:

z = (p̂ - p) / √(p * (1 - p) / n)

Where:
p̂ - sample proportion (0.28)
p - assumed population proportion (0.26)
n - sample size (700)

Substituting the values:

z = (0.28 - 0.26) / √(0.26 * (1 - 0.26) / 700)
z = 0.02 / √(0.26 * 0.74 / 700)
z ≈ 0.02 / 0.0158
z ≈ 1.27

Now, we need to find the P-value associated with the z-score of 1.27. Looking up in a standard normal distribution table or using a calculator/statistical software, we find that the P-value for this test is approximately 0.102.

Based on the P-value, which is greater than the significance level (typically 0.05 or 0.01), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the percentage of businesses owned by women has increased.