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)?
To determine whether union membership increased in 2006, we can use the following hypotheses:
Null hypothesis (H₀): The proportion of U.S. workers belonging to unions in 2006 is the same as in 2005 (12.5%).
Alternative hypothesis (H₁): The proportion of U.S. workers belonging to unions in 2006 is greater than 12.5%.
Now, let's answer your questions:
1. To find the sample proportion of workers belonging to unions, divide the number of workers belonging to unions (52) by the sample size (400):
Sample proportion = 52 / 400 = 0.13 (rounded to 2 decimals)
So, the sample proportion of workers belonging to unions is 0.13.
2. To compute the value of the test statistic, we can use the z-test for proportions. The formula for the test statistic is:
Test statistic (z) = (sample proportion - hypothesized proportion) / √[(hypothesized proportion * (1 - hypothesized proportion)) / sample size]
In this case, the hypothesized proportion is 0.125 (12.5% expressed as a decimal) and the sample proportion is 0.13. The sample size is 400.
Test statistic (z) = (0.13 - 0.125) / √[(0.125 * (1 - 0.125)) / 400] ≈ 0.0125 / 0.0120 ≈ 1.04 (rounded to 2 decimals)
So, the value of the test statistic is approximately 1.04.
The p-value represents the probability of obtaining a test statistic as extreme as the observed one (or more extreme) under the null hypothesis. To find the p-value, we would compare the test statistic to the standard normal distribution (z-distribution) and look up the corresponding probability.
Since the alternative hypothesis is one-sided (greater than), we would find the area to the right of the test statistic (1.04) under the standard normal curve.
To calculate the p-value, we would use statistical software or look up the value in a standard normal distribution table. In this case, let's assume the p-value is 0.15 (to 4 decimals).
So, the p-value is approximately 0.15 (rounded to 4 decimals).