The mean annual incomes of certified welders are normally distributed with the mean of $50,000 and a population standard deviation of $2,000. The ship building association wishes to find out whether their welders earn more or less than $50,000 annually. A sample of 100 welders is taken and the mean annual income of the sample is $50,350. If the level of significance is 0.10,what conclusion should be drawn?
A)Do not reject the null hypothesis as the test statistic is less than the critical value of z.
B)Do not reject the null hypothesis as the test statistic is less than the critical value of t.
C)Reject the null hypothesis as the test statistic is greater than the critical value of t.
D)Reject the null hypothesis as the test statistic is greater than the critical value of z.

Question

Quizplus · Accepted Answer

We need to test the hypothesis if the welders earn more or less than $50,000 annually. Let us assume the null hypothesis and the alternate hypothesis as follows:
Null Hypothesis (H0): μ ≤ $50,000 (The mean income of welders is less than or equal to $50,000 per annum)
Alternate Hypothesis (Ha): μ > $50,000 (The mean income of welders is greater than $50,000 per annum)

The sample size is 100, which is greater than 30. As the population standard deviation is known, we can use the z-test for the mean. The level of significance is 0.10, which means that the critical value of z at 10% level of significance is 1.28 (from the standard normal distribution table).

The test statistic can be calculated as follows:
z = (x̄ - μ) / (σ / sqrt(n))
z = (50,350 - 50,000) / (2000 / sqrt(100))
z = 3.5

Since the calculated z-value is greater than the critical value of z (3.5 > 1.28), we reject the null hypothesis. Hence, we can conclude that the welders earn more than $50,000 annually with 90% confidence.

The Mean Annual Incomes of Certified Welders Are Normally Distributed