The following data set represents the repair costs (in dollars) for a random sample of 30 dishwashers.
41.82 52.81 57.80 68.16 73.48 78.88 88.13 88.79
90.07 90.35 91.68 91.72 93.01 95.21 95.34 96.50
100.05 101.32 103.59 104.19 105.62 111.32 117.14 118.42
118.77 119.01 120.70 140.52 141.84 147.06
(a) Find the point estimate of the population mean.
(b) Find the margin of error for a 95% level of confidence.
(c) Construct a 95% confidence interval for the population mean and interpret the results.
The point of estimate is 97.11. Is that correct?
I don't know where to start with b
To find the point estimate of the population mean, you need to calculate the sample mean. The sample mean is the sum of all the data points divided by the total number of data points in the sample.
In this case, you have a sample of 30 dishwasher repair costs. Add up all the values in the dataset:
41.82 + 52.81 + 57.80 + 68.16 + 73.48 + 78.88 + 88.13 + 88.79 + 90.07 + 90.35 + 91.68 + 91.72 + 93.01 + 95.21 + 95.34 + 96.50 + 100.05 + 101.32 + 103.59 + 104.19 + 105.62 + 111.32 + 117.14 + 118.42 + 118.77 + 119.01 + 120.70 + 140.52 + 141.84 + 147.06 = 2,913.74
Divide this sum by the total number of data points (30):
2,913.74 / 30 = 97.125
Therefore, the point estimate of the population mean is approximately 97.125 (rounded to three decimal places).
For part (b), to find the margin of error for a 95% level of confidence, you will need to use the formula for the margin of error, which is given by:
Margin of Error = Critical Value * Standard Deviation / Square Root of Sample Size.
Since you have a small sample size (30) and the population standard deviation is unknown, you will need to use the t-distribution and find the critical value from a t-table.
To find the critical value, you need to know the degrees of freedom, which is calculated as (n-1), where n is the sample size. In this case, the degrees of freedom is (30-1) = 29.
For a 95% confidence level with 29 degrees of freedom, the critical value from the t-table is approximately 2.045.
The next step is to calculate the sample standard deviation, denoted as "s." To find the sample standard deviation, first, calculate the sum of squared differences between each data point and the sample mean:
(41.82 - 97.125)^2 +
(52.81 - 97.125)^2 +
(57.80 - 97.125)^2 +
(68.16 - 97.125)^2 +
(73.48 - 97.125)^2 +
(78.88 - 97.125)^2 +
(88.13 - 97.125)^2 +
(88.79 - 97.125)^2 +
(90.07 - 97.125)^2 +
(90.35 - 97.125)^2 +
(91.68 - 97.125)^2 +
(91.72 - 97.125)^2 +
(93.01 - 97.125)^2 +
(95.21 - 97.125)^2 +
(95.34 - 97.125)^2 +
(96.50 - 97.125)^2 +
(100.05 - 97.125)^2 +
(101.32 - 97.125)^2 +
(103.59 - 97.125)^2 +
(104.19 - 97.125)^2 +
(105.62 - 97.125)^2 +
(111.32 - 97.125)^2 +
(117.14 - 97.125)^2 +
(118.42 - 97.125)^2 +
(118.77 - 97.125)^2 +
(119.01 - 97.125)^2 +
(120.70 - 97.125)^2 +
(140.52 - 97.125)^2 +
(141.84 - 97.125)^2 +
(147.06 - 97.125)^2
Calculate the sum of squared differences:
17,373.928775
Divide the sum of squared differences by (n-1):
17,373.928775 / (30-1) = 615.497813
Take the square root of the result to find the sample standard deviation (s):
√615.497813 ≈ 24.792
Now you have all the values needed to find the margin of error. Using the formula:
Margin of Error = Critical Value * Standard Deviation / Square Root of Sample Size
Margin of Error = 2.045 * 24.792 / √30
Margin of Error ≈ 8.308
Therefore, the margin of error for a 95% level of confidence is approximately 8.308 (rounded to three decimal places).
For part (c), to construct a 95% confidence interval for the population mean, you will use the point estimate (sample mean) and the margin of error.
The confidence interval is given by:
Confidence Interval = Point Estimate ± Margin of Error
Substituting the values:
Confidence Interval = 97.125 ± 8.308
Confidence Interval ≈ (88.817, 105.433)
Interpretation of the results: We are 95% confident that the true population mean repair cost for dishwashers falls within the interval of approximately $88.817 to $105.433. This means that if we were to take many random samples and calculate confidence intervals for each, approximately 95% of those intervals would contain the true population mean repair cost.