A company has two factories in which they manufacture engines. Once a month they randomly select 10 engines from each factory and test if there is a difference in performance in engines made in the two factories. This month the average output of the motors from Factory 1 is 120 horsepower with a standard deviation of 5 horsepower, and the average output of the motors from Factory 2 is 132 horsepower with a standard deviation of 4 horsepower.
(a) Calculate a 95% confidence interval for the difference in the average horsepower for engines coming from the two factories and interpret it in context.
(b) Based on your confidence interval, is there a significant evidence that there is a dif- ference in performance in engines made in the two factories? If so, can you tell which factory produces motors with lower performance? Explain.
(c) Recently upgrades were made in Factory 2. Do these data prove that these upgrades enhanced the performance in engines made in this factory? Explain.
You'll need to start by finding a 95% confidence interval for 2 samples. If you use a z-table, then here is an example of a formula:
CI95 = (x1 - x2) ± 1.96 [√(s1^2/n1 + s2^2/n2)]
...where x1, x2 are the means; ± 1.96 represents the 95% confidence interval using a z-table; s1, s2 are the standard deviations (^2 means squared); and n1, n2 are the sample sizes.
Plug the values from your problem into the formula and determine the interval.
I hope this brief explanation will help get you started.
To calculate a 95% confidence interval for the difference in the average horsepower for engines coming from the two factories, we can use the two-sample t-test. Here's how:
Step 1: Define your hypotheses:
- Null hypothesis (H0): The average horsepower for engines from Factory 1 is the same as the average horsepower for engines from Factory 2.
- Alternative hypothesis (Ha): The average horsepower for engines from Factory 1 is different from the average horsepower for engines from Factory 2.
Step 2: Calculate the pooled standard error:
The pooled standard error is a measure of the combined variability in the two samples. It can be calculated using the following formula:
Pooled Standard Error = sqrt((s1^2/n1) + (s2^2/n2))
where s1 and s2 are the standard deviations of Factory 1 and Factory 2 respectively, n1 and n2 are the sample sizes.
In this case, s1 = 5, s2 = 4, n1 = 10, and n2 = 10.
Pooled Standard Error = sqrt((5^2/10) + (4^2/10)) = sqrt(2.5 + 1.6) = sqrt(4.1) ≈ 2.02
Step 3: Calculate the t-value:
The t-value represents the number of standard errors that the sample mean is away from the null hypothesis. It can be calculated using the following formula:
t = (x̄1 - x̄2) / Pooled Standard Error
where x̄1 and x̄2 are the sample means of Factory 1 and Factory 2 respectively.
In this case, x̄1 = 120, x̄2 = 132.
t = (120 - 132) / 2.02 ≈ -5.94
Step 4: Calculate the degrees of freedom:
The degrees of freedom can be calculated using the following formula:
df = (n1 - 1) + (n2 - 1)
In this case, df = (10 - 1) + (10 - 1) = 18.
Step 5: Look up the critical t-value and calculate the confidence interval:
Using a t-table or a t-distribution calculator with the degrees of freedom and desired confidence level (95% in this case), we find the critical t-value to be approximately ±2.101.
Confidence Interval = (x̄1 - x̄2) ± (Critical t-value) * Pooled Standard Error
Confidence Interval = (120 - 132) ± 2.101 * 2.02 ≈ -12 ± 4.23
So the 95% confidence interval for the difference in average horsepower for engines from the two factories is approximately -16.23 to -7.77.
(a) Interpreting the confidence interval in context:
We can say with 95% confidence that the true average difference in horsepower between engines from Factory 1 and Factory 2 lies between -16.23 and -7.77 horsepower. Since the interval does not include zero, we can interpret that there is a statistically significant difference in performance between the two factories.
(b) Based on the confidence interval:
Since the confidence interval lies entirely in negative values, we can conclude that Factory 2 produces motors with lower performance compared to Factory 1.
(c) Evaluating the upgrades:
We cannot conclude from these data alone that the upgrades in Factory 2 have enhanced the performance of engines made there. While there is evidence of a difference in performance between the two factories, this does not explicitly indicate the cause. Other factors could be contributing to the performance difference, and further analysis or experiments would be required to attribute it solely to the upgrades made in Factory 2.