Question

a student records the repair cost for 27 randomly selected dryers. a sample mean of $58.03 and standard deviation of $26.34 are subsequently computed. determine the 90% confidence interval for the mean repair cost for the dryers. assume the population is approximately normal. step 1 of 2 : find the critical value that should be used in constructing the confidence interval. round your answer to three decimal places.

Answer 1

To find the critical value for a 90% confidence interval, you need to determine the Z-score for that confidence level. The Z-score represents the number of standard deviations away from the mean that corresponds to a certain confidence level. For a 90% confidence level, the Z-score is approximately 1.645.

Step-by-step explanation:

To find the critical value, we need to look up the Z-score for a 90% confidence level. The Z-score is the number of standard deviations away from the mean that corresponds to a specific confidence level. In this case, we want to find the Z-score that corresponds to a 90% confidence level.

Using a standard normal distribution table or a Z-score calculator, we find that the Z-score for a 90% confidence level is approximately 1.645. This means that approximately 90% of the area under the standard normal curve falls within 1.645 standard deviations from the mean.

Answer 2

To find the critical value for a 90% confidence interval, we use the z-score corresponding to the desired confidence level. For a 90% confidence level, the critical value is approximately ($51.613, $64.447).

Step-by-step explanation:

To find the 90% confidence interval for the mean repair cost for the dryers, we first need to determine the critical value that should be used in constructing the confidence interval. We’ll use the standard normal distribution (Z-distribution) for this calculation.

Step 1 of 2: Find the critical value that should be used in constructing the confidence interval.

We can use the following formula to find the critical value (z) for a 90% confidence interval:

z = Zα/2

where Zα/2 is the critical value from the standard normal distribution table that corresponds to α/2 (which is 0.05/2 = 0.025 for a 90% confidence interval).

Using a standard normal distribution table or calculator, we find that the critical value (z) for a 90% confidence interval is approximately 1.645.

Step 2 of 2: Calculate the margin of error and the confidence interval.

To calculate the margin of error (ME), we use the following formula:

ME = z * (standard deviation / √n)

where z is the critical value found in Step 1, the standard deviation is $26.34, and n is the sample size (27 dryers).

ME = 1.645 * (26.34 / √27) ≈ 6.417

Now, we can calculate the 90% confidence interval for the mean repair cost for the dryers:

Lower limit = sample mean - ME = $58.03 - $6.417 ≈ $51.613 Upper limit = sample mean + ME = $58.03 + $6.417 ≈ $64.447

So, the 90% confidence interval for the mean repair cost for the dryers is approximately ($51.613, $64.447).