The average annual salary of employees at Wintertime Sports was $28,750 last year. This year the company opened another store. Suppose a random sample of 18 employees had an average annual salary of $25,810 with standard deviation of s = $4230. Use a 1% level of significance to test the claim that the average annual salary for all employees is different from last year's average salary.
To test the claim that the average annual salary for all employees is different from last year's average salary, we can perform a two-sample t-test.
Here are the steps to perform the test:
Step 1: State the null and alternative hypotheses.
Null hypothesis (H0): The average annual salary for all employees is the same as last year's average salary.
Alternative hypothesis (Ha): The average annual salary for all employees is different from last year's average salary.
Step 2: Determine the significance level.
The significance level is given as 1% or 0.01.
Step 3: Calculate the test statistic.
For a two-sample t-test, the test statistic is given by:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where:
x1 = sample mean of the new store employees
x2 = population mean of the previous year's employees
s1 = standard deviation of the new store employees
n1 = sample size of the new store employees
n2 = sample size of the previous year's employees
In this case, x1 = $25,810, x2 = $28,750, s1 = $4230, n1 = 18, and n2 is not provided.
Step 4: Determine the critical value(s).
Since the alternative hypothesis is two-sided (different from), we need to find the critical values for a two-sided test at the given significance level. With a sample size of 18 (n1 = n2 = 18), we can consult the t-distribution table or use statistical software to find the critical values.
Assuming a normal distribution and using a 1% significance level, we can find a t-critical value of approximately ±2.898.
Step 5: Make a decision.
If the absolute value of the test statistic t is greater than the critical value (t > 2.898 or t < -2.898), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value.
The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. If the p-value is less than the significance level (0.01), we reject the null hypothesis.
You would need to calculate the test statistic from Step 3, compare it to the critical value from Step 4, and calculate the p-value to make a decision on whether to reject or fail to reject the null hypothesis.
Note: The sample sizes for both years should ideally be the same for a proper comparison. If the sample size for the previous year's employees is also given, it should be used to calculate the test statistic and critical value.
To test the claim that the average annual salary for all employees is different from last year's average salary, we can use a t-test.
Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The average annual salary for all employees is the same as last year's average salary. µ = $28,750
Alternative hypothesis (Ha): The average annual salary for all employees is different from last year's average salary. µ ≠ $28,750
Step 2: Determine the level of significance:
The given level of significance is 1%, which corresponds to a significance level (α) of 0.01.
Step 3: Compute the test statistic:
The formula for the t-test statistic is:
t = (x̄ - µ) / (s / √n)
Where:
x̄ = sample mean = $25,810
µ = population mean (last year's average salary) = $28,750
s = sample standard deviation = $4230
n = sample size = 18
Plugging in the values, we get:
t = (25810 - 28750) / (4230 / √18)
Step 4: Find the critical value(s):
Since we are using a two-tailed test, the critical values will be determined using a t-distribution with (n-1) degrees of freedom (df = 18-1 = 17). The critical value can be obtained from a t-distribution table or a statistical software. For a 1% level of significance (α = 0.01), the critical value is approximately ±2.898.
Step 5: Make a decision:
If the absolute value of the calculated test statistic (|t|) is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the test statistic and compare with the critical value:
t = (25810 - 28750) / (4230 / √18)
= -3479.899 / (4230 / √18)
≈ -12.02
Since the absolute value of the test statistic (|t|) is greater than the critical value (|t| > 2.898), we reject the null hypothesis.
Step 7: Interpret the result:
Based on the given sample data, we have enough evidence to suggest that the average annual salary for all employees this year is different from last year's average salary.