Wages and salaries make up only part of a total compensation. Other parts include paid leave, health insurance, and many others. In 2007, wages and salaries among manufacturers in the United States made up an average of 65.8% of total compensation. To determine if this changed in 2008 a random sample of manufacturing employees was drawn. Can we infer that percentage of total compensation for wages and salaries increased between 2007 and 2008?
Sample data lacking.
To determine if the percentage of total compensation for wages and salaries increased between 2007 and 2008, we can use hypothesis testing. Here is how you can go about it:
1. State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The percentage of total compensation for wages and salaries in 2008 is the same as in 2007.
- Alternative hypothesis (Ha): The percentage of total compensation for wages and salaries in 2008 is different from that in 2007.
2. Set the significance level (α) for your hypothesis test. A common choice is α = 0.05, which corresponds to a 5% chance of rejecting the null hypothesis when it is true.
3. Calculate the sample statistics:
- Calculate the sample mean and standard deviation for the percentage of total compensation for wages and salaries in the random sample of manufacturing employees in 2008.
4. Conduct the hypothesis test:
- Perform a t-test to compare the sample mean from 2008 with the known population proportion from 2007.
- Calculate the test statistic, which measures the difference between the sample mean and the population proportion under the null hypothesis.
5. Determine the critical region:
- Based on the significance level (α) and the degrees of freedom of the t-distribution, determine the critical region of the test statistic.
- If the test statistic falls within the critical region, reject the null hypothesis.
6. Calculate the p-value:
- Calculate the p-value associated with the observed test statistic.
- The p-value indicates the probability of observing a test statistic as extreme as the one calculated under the null hypothesis.
7. Make a decision:
- If the p-value is less than the significance level (α), reject the null hypothesis.
- If the p-value is greater than the significance level (α), fail to reject the null hypothesis.
Keep in mind that statistical inference requires some assumptions, such as a random sample and the approximate normal distribution of the population. Remember to check the conditions for applying the t-test before proceeding with the analysis.