Question

A manufacturer knows that the numbers of items produced per hour by machine A and by machine B are normally distributed with a standard deviation of 8.4 items for machine A and a standard deviation of 11.3 items for machine B. The mean hourly amount produced by machine A for a random sample of 40 hours was 130 items; the mean hourly amount produced by machine B for a random sample of 36 hours was 120 items.

a. Find the 95% confidence interval for the difference in mean items produced per hour by these two machines.

b. Conduct a test if machine A produces more than items than machine B per hour. Use the critical value approach with the 5% significance level.

c. Compute the p-value for the test in part b and draw your conclusion.

Answer 1

The 95% confidence interval for the difference in mean items produced per hour between machine A and B is calculated using standard error and Z-score. A hypothesis test is conducted to see if machine A produces more than machine B, rejecting the null hypothesis if the Z value is greater than 1.96. The p-value helps to determine the strength of evidence against the null hypothesis.

Step-by-step explanation:

95% Confidence Interval

To find the 95% confidence interval for the difference in mean items produced by machines A and B, we need to use the formula for the confidence interval of the difference between two independent sample means:

CI = (μ1 - μ2) ± Z* √((σ²/n1) + (σ²/n2))

Where μ1 and μ2 are the sample means, σ² are the variances (standard deviations squared), n1 and n2 are the sample sizes, and Z* is the Z-score corresponding to the desired confidence level. Here, the sample means are 130 and 120 items respectively, the standard deviations are 8.4 and 11.3 items respectively, and the sample sizes are 40 and 36 hours respectively.

First, calculate the standard error (SE) of the difference between the two means:

• 
  
• SE = √((8.4²/40) + (11.3²/36))

The Z-score for a 95% confidence level is approximately 1.96, so we can then calculate the margin of error (ME):

• 
  
• ME = 1.96 * SE

The confidence interval (Δμ) is then:

• 
  
• Δμ = (130 - 120) ± ME

To conduct the test if machine A produces more items than machine B per hour, we perform a hypothesis test for the difference between means. We set up the null hypothesis (H0: μA - μB <= 0) and the alternative hypothesis (H1: μA - μB > 0).

We then calculate the test statistic using the following formula:

• 
  
• Z = (μ1 - μ2) / SE

If Z is greater than the critical value of Z (1.96 for a 5% significance level, one-tailed test), we reject the null hypothesis.

For the p-value, it is calculated as the probability of observing a test statistic as extreme as, or more extreme than, the one observed under the assumption that the null hypothesis is true. We compare the p-value with the significance level to decide whether to reject the null hypothesis.

A small p-value (typically <= 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.