A reseacher would like to determine whether a change in lighting to full spectrum light bulbs increases productivity on an assembly line. A sample of n= 36 participants is obtained, and the light bulbs over the line are replaced. under general/historical operating procedures, productivity has a mean of 330 units per day and a standard deviation of 10. The individuals in the sample had an average productivity of 333 units and a s=11. A)Can the research team conclude that the change of light bulbs increased productivity with p level of 0.01? show all necessary components to support your answer when testing this hypothesis. B) The distribution of units produced is not normally distributed in the population. is it still all right to do a hypothesis test? Explain your answer.
A) I would say the research team cannot conclude that the change of light bulbs increased productivity with the data given. However, check this by using the appropriate hypothesis test.
B) Yes, you can still do a hypothesis test by using one of the tenets of the Central Limit Theorem:
The shape of the sampling distribution increasingly approximates a normal curve as sample size (n) is increased, even if the original population is not normally distributed.
I hope this brief explanation will help.
A) To determine if the change in lighting to full spectrum light bulbs increased productivity, the research team can perform a hypothesis test using the given data. Here are the necessary components to support the answer:
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H0): The change in light bulbs does not increase productivity (μ = 330).
- Alternative Hypothesis (Ha): The change in light bulbs increases productivity (μ > 330).
Step 2: Choose the appropriate statistical test and significance level.
Since the sample size (n = 36) is greater than 30 and the population standard deviation (σ) is known, we can use a z-test. The significance level given is α = 0.01.
Step 3: Compute the test statistic.
Calculate the z-score using the formula:
z = (x̄ - μ) / (σ / √n)
where x̄ is the sample mean (333), μ is the population mean (330), σ is the population standard deviation (10), and n is the sample size (36).
Substituting the values:
z = (333 - 330) / (10 / √36) = 3 / (10 / 6) = 1.8
Step 4: Determine the critical value.
Since the alternative hypothesis is one-sided (μ > 330), we need to find the critical value for a right-tailed test at a 0.01 significance level. By referring to the z-table or using a calculator, the critical value for α = 0.01 is approximately 2.326.
Step 5: Make a decision and interpret the result.
Compare the calculated test statistic (1.8) with the critical value (2.326). Since the test statistic is smaller than the critical value (1.8 < 2.326), we fail to reject the null hypothesis. Therefore, the research team cannot conclude that the change in light bulbs significantly increased productivity at a 0.01 significance level.
B) If the distribution of units produced is not normally distributed in the population, it might affect the validity of the hypothesis test. The assumption of normality is necessary for accurate inference in many statistical tests. However, in large sample sizes (n > 30), the Central Limit Theorem states that the sampling distribution will tend to be normally distributed regardless of the population distribution. Therefore, the hypothesis test can still be performed if the sample size is large enough, despite the population not being normally distributed.