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)?
1. To find the sample proportion of workers belonging to unions, divide the number of workers belonging to unions by the total sample size:
Sample proportion = (Number of workers belonging to unions) / (Total sample size)
Sample proportion = 52 / 400
Sample proportion ≈ 0.13 (rounded to 2 decimals)
2. To test whether union membership increased in 2006, we can set up the following hypotheses:
Null hypothesis (H0): The proportion of workers belonging to unions in 2006 is equal to or less than the proportion in 2005.
Alternative hypothesis (Ha): The proportion of workers belonging to unions in 2006 is greater than the proportion in 2005.
Since we are testing whether the proportion increased, it is a one-tailed test with the alternative hypothesis stating that the proportion is greater.
To compute the value of the test statistic, we can use the formula for the z-test for proportions:
z = (Sample proportion - Population proportion) / sqrt((Population proportion * (1 - Population proportion)) / Sample size)
Here, the population proportion is the proportion in 2005 (12.5% or 0.125), and the sample size is 400.
Substituting the values:
z = (0.13 - 0.125) / sqrt((0.125 * (1 - 0.125)) / 400)
z ≈ 0.005 / sqrt(0.11015625 / 400)
z ≈ 0.005 / sqrt(0.00027539)
z ≈ 0.005 / 0.016587
z ≈ 0.3018 (rounded to 2 decimals)
To find the p-value, we can use a standard normal distribution table or a statistical calculator. Since the alternative hypothesis is one-tailed (proportion is greater), we need to find the area under the curve to the right of the test statistic.
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the observed value, assuming the null hypothesis is true.
Using a standard normal distribution table or calculator, the p-value for z = 0.3018 is approximately 0.3829 (rounded to 4 decimals).
Therefore, the p-value is approximately 0.3829.
To determine whether union membership increased in 2006, we can use hypothesis testing.
1. The null hypothesis (H0) states that there was no increase in union membership in 2006. The alternative hypothesis (Ha) states that there was an increase in union membership.
H0: p = 0.125 (proportion of workers belonging to unions in 2005)
Ha: p > 0.125
2. To calculate the sample proportion of workers belonging to unions, you divide the number of workers belonging to unions (52) by the total sample size (400).
Sample proportion = 52/400 = 0.13 (to 2 decimals)
3. To compute the test statistic, we need to calculate the z-score. The formula for calculating the z-score is z = (sample proportion - null proportion) / standard error. The standard error is the square root of [(null proportion * (1 - null proportion)) / sample size].
Null proportion = 0.125
Sample size = 400
Standard error = sqrt((0.125 * 0.875) / 400)
z = (0.13 - 0.125) / sqrt((0.125 * 0.875) / 400) = 0.005 / 0.0118 = 0.4237 (to 2 decimals)
4. The p-value is the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true. To calculate the p-value, we need to find the area under the normal distribution curve to the right of the calculated z-score. You can use a z-table or a statistical software to find the p-value.
For a z-score of 0.42, the p-value is approximately 0.3355 (to 4 decimals).
Therefore, the p-value is approximately 0.3355 (to 4 decimals).