The data below represent a random sample of 15 defective items produced by a production process they are:

General
1

The data below represent a random sample of 15 defective items produced by a production process they are:
500, 210,420,610,450,253,187,480,310,383,520,428,530,280,240 (Units).
Construct 95% and 99% confident interval for the average weekly defective items that passed through the production process.

2

Construct 95% and 99% confident interval for the average weekly defective items that passed through the production process.

To construct the confidence intervals, we need to follow these steps:

  1. Calculate the sample mean (\bar{x}).
  2. Calculate the sample standard deviation (s).
  3. Determine the critical t-values for 95% and 99% confidence levels.
  4. Use the formula for the confidence interval.

Let’s start with our data set of 15 defective items: (500, 210, 420, 610, 450, 253, 187, 480, 310, 383, 520, 428, 530, 280, 240).

Step 1: Calculate the Sample Mean (\bar{x}):

\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Where:

  • ( n ) is the sample size (15 in this case).
  • ( x_i ) is each individual sample value.

Sum up all the values:

500 + 210 + 420 + 610 + 450 + 253 + 187 + 480 + 310 + 383 + 520 + 428 + 530 + 280 + 240 = 5801

The sample mean is:

\bar{x} = \frac{5801}{15} \approx 386.73

Step 2: Calculate the Sample Standard Deviation (s):

First, find the squared differences between each data point and the mean, sum them up, and then divide by ( n-1 ):

s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}

Calculate each squared difference:

(500-386.73)^2 + (210-386.73)^2 + \ldots + (240-386.73)^2

Sum these squared differences, then divide by 14 (since ( n-1 = 15-1 )):

s \approx \sqrt{\frac{(128.33)^2 + (-176.73)^2 + \ldots + (-146.73)^2}{14}} \approx 125.35

Step 3: Determine the Critical t-Values for 95% and 99% Confidence Levels:

For a sample size of 15 (degrees of freedom = 14):

  • The critical t-value for a 95% confidence level: t_{0.025, 14} \approx 2.145
  • The critical t-value for a 99% confidence level: t_{0.005, 14} \approx 2.977

Step 4: Calculate the Confidence Intervals:

The formula for the confidence interval is:

\bar{x} \pm t_{\alpha/2, df} \left(\frac{s}{\sqrt{n}}\right)

95% Confidence Interval:

386.73 \pm 2.145 \left(\frac{125.35}{\sqrt{15}}\right)
386.73 \pm 2.145 \left(\frac{125.35}{3.872}\right)
386.73 \pm 69.43

Thus, the 95% confidence interval is:

[317.30, 456.16]

99% Confidence Interval:

386.73 \pm 2.977 \left(\frac{125.35}{\sqrt{15}}\right)
386.73 \pm 2.977 \left(\frac{125.35}{3.872}\right)
386.73 \pm 96.5

Thus, the 99% confidence interval is:

[290.23, 483.23]

Conclusion:

  • 95% Confidence Interval for the average weekly defective items: [317.30, 456.16]
  • 99% Confidence Interval for the average weekly defective items: [290.23, 483.23]

These intervals provide ranges within which we can be 95% or 99% confident that the true mean of defective items produced by the process lies.