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 hypothesis testing.
Null hypothesis (H0): Union membership in 2006 is the same as in 2005.
Alternative hypothesis (Ha): Union membership in 2006 has increased compared to 2005.
1. To calculate the sample proportion, divide the number of workers belonging to unions (52) by the total sample size (400).
Sample proportion = 52/400 = 0.13 (rounded to 2 decimals)
2. To compute the test statistic, we can use the formula for a hypothesis test for proportions:
Test statistic = (Sample proportion - Hypothesized proportion) / Standard error
Since the null hypothesis states that union membership in 2006 is the same as in 2005, the hypothesized proportion is 0.125 (12.5% expressed as a decimal).
Standard error = sqrt[(Hypothesized proportion * (1 - Hypothesized proportion)) / Sample size]
Standard error = sqrt[(0.125 * (1 - 0.125)) / 400] = sqrt(0.0003575) ≈ 0.0189 (rounded to 2 decimals)
Test statistic = (0.13 - 0.125) / 0.0189 = 0.0053 / 0.0189 ≈ 0.28 (rounded to 2 decimals)
To find the p-value, we need to compare the test statistic to the appropriate distribution. In this case, we can use a Z-distribution.
First, determine the critical value(s) based on the significance level (α) of 0.05. This is typically done using a Z-table or calculator.
Then, find the area under the Z-distribution curve beyond the test statistic in the direction of the alternative hypothesis. This will give us the p-value.
Since the test statistic is very small (0.28), it is unlikely to be significant. However, the actual p-value calculation requires knowing the critical value(s) and comparing them to the test statistic.
Without the critical value(s), it is not possible to determine the exact p-value in this case.