1. A manager wishes to find out whether there is a relationship between the age of his employees and the number of sick days they take each year. The data is given below:
Age (X) 18 26 39 48 53 58 24 50
Days (Y) 16 12 9 5 6 2 10 7
a. What is the null hypothesis?
b. What is the strength and the direction of relationship?
c. Is the relationship significant?
d. What proportion of the number of sick days can be explained by the age of the employee?
e. In a sentence or so, give an appropriate interpretation of the analysis results.
Use the Pearson r.
To find if there is a statistically significant linear relationship between hours worked and error rate, use N-2 for degrees of freedom at the appropriate significance level for the test you are doing. Use a table for critical or cutoff values for a Pearson r. Compare the value from the table to the value you calculated using a formula for the Pearson r. If the value you calculate exceeds the critical value from the table, the null will be rejected. There will be a linear relationship in the population and the test will be statistically significant. If the value you calculate does not exceed the critical value from the table, then the null will not be rejected and you cannot conclude a linear relationship in the population.
I hope this brief explanation will get you started.
Correction - the first sentence should read as follows:
To find if there is a statistically significant linear relationship between age and number of sick days, use N-2 for degrees of freedom at the appropriate significance level for the test you are doing.
Sorry for any confusion.
a. The null hypothesis is that there is no relationship between the age of employees and the number of sick days they take each year.
b. The strength and direction of relationship can be determined by calculating the correlation coefficient. In this case, it would require further calculation using statistical software or formulae.
c. To determine if the relationship is significant, a hypothesis test can be performed. This could be done using techniques such as regression analysis or calculating the p-value.
d. The proportion of the number of sick days that can be explained by the age of the employee can be determined using the coefficient of determination (R-squared) in regression analysis.
e. An appropriate interpretation of the analysis results would depend on the specific statistical analysis performed and the calculated values. However, it would generally involve stating whether there is a significant relationship between age and sick days, the direction and strength of the relationship, and the proportion of variability explained by age.
a. The null hypothesis in this case would state that there is no relationship between the age of the employees and the number of sick days they take each year.
b. To determine the strength and direction of the relationship, we can calculate the correlation coefficient. The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, we need to calculate the correlation coefficient between age (X) and sick days (Y).
c. To determine if the relationship is significant, we can perform a hypothesis test. One common test is the Pearson's correlation test. In this test, we determine the probability of observing a correlation coefficient as extreme as the one calculated if the null hypothesis were true. If the probability (also called p-value) is below a certain threshold (such as 0.05), we reject the null hypothesis and conclude that there is a statistically significant relationship.
d. To determine the proportion of the number of sick days that can be explained by the age of the employee, we can calculate the coefficient of determination (R-squared). R-squared represents the proportion of the dependent variable's (sick days) variation that can be explained by the independent variable (age). It ranges from 0 to 1, where 0 means no variation is explained and 1 means all the variation is explained.
e. Based on the analysis results, we can interpret whether there is a relationship between the age of the employees and the number of sick days they take each year. If the null hypothesis is rejected and the correlation coefficient is significantly different from zero, we can conclude that there is a statistically significant relationship between age and sick days. Additionally, the proportion of the number of sick days that can be explained by the age of the employee can provide insights into how much age influences sick days taken.