A study found that, in 2005, 12.5% of U.S. workers belonged to unions (The Wall Street Journal, January 21, 2006). Suppose a sample of 400 U.S. workers is collected in 2006 to determine whether union efforts to organize have increased union membership.
Formulate the hypotheses that can be used to determine whether union membership increased in 2006.
1. If the sample results show that 52 of the workers belonged to unions, what is the sample proportion of workers belonging to unions (to 2 decimals)?
2. Complete the following, assuming an level of .05.
Compute the value of the test statistic (to 2 decimals).
What is the p-value (to 4 decimals)?
.3258
To determine whether union membership has increased in 2006, we can set up the following hypotheses:
Null hypothesis (H0): The proportion of U.S. workers belonging to unions in 2006 is the same as in 2005.
Alternative hypothesis (Ha): The proportion of U.S. workers belonging to unions in 2006 is greater than in 2005.
Now let's answer the specific questions:
1. To determine the sample proportion of workers belonging to unions, we divide the number of workers belonging to unions (52) by the total sample size (400). In this case, the sample proportion is 52/400 = 0.13 (rounded to 2 decimal places).
2. Assuming a significance level of 0.05, we can use a one-sample proportion test to determine the value of the test statistic and the p-value.
To compute the test statistic:
- First, calculate the standard error of the proportion using the formula:
SE = sqrt[(p * (1 - p)) / n]
Where p is the hypothesized population proportion and n is the sample size.
- In this case, the hypothesized population proportion is given by the 2005 union membership rate of 12.5%, which in decimal form is 0.125. Thus:
SE = sqrt[(0.125 * (1 - 0.125)) / 400] = 0.015
- Next, calculate the test statistic (Z-score) using the formula:
Z = (p - p0) / SE
Where p is the sample proportion and p0 is the hypothesized population proportion.
- In this case, p = 0.13 (sample proportion) and p0 = 0.125 (hypothesized population proportion). Thus:
Z = (0.13 - 0.125) / 0.015 = 0.333
The test statistic (Z-score) is 0.333.
To find the p-value associated with this test statistic, we compare it to the standard normal distribution. Since the alternative hypothesis is that the proportion has increased, we are conducting a one-tailed test.
Using the standard normal distribution table or a statistical calculator, we find that the p-value for a Z-score of 0.333 is approximately 0.6274 (rounded to 4 decimal places).
Therefore, the p-value is 0.6274.