document.write( "Question 1170777: You are given the sample mean and the sample standard deviation Assume the population is normally distributed and use the T distribution define margine of error and construct a 95% confidence interval for the population mean. Interpret the results.\r
\n" ); document.write( "\n" ); document.write( "1. In a random sample of 8 people, the main commute time to work was 35.5 minutes and the standard deviation was 7.2 minutes.\r
\n" ); document.write( "\n" ); document.write( "2. In a random sample of 13 microwaves, the main repair cost was $80.00 and the standard deviation was $13.50.\r
\n" ); document.write( "\n" ); document.write( "3. In a random sample of 7 computers, the repair cost was $110.00 and a standard division was $44.50
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\n" ); document.write( "\n" ); document.write( "**General Formula for Confidence Interval**\r
\n" ); document.write( "\n" ); document.write( "When the population standard deviation is unknown and the sample size is small (n < 30), we use the t-distribution.\r
\n" ); document.write( "\n" ); document.write( "Confidence Interval = Sample Mean ± Margin of Error\r
\n" ); document.write( "\n" ); document.write( "Margin of Error (E) = t_c * (s / √n)\r
\n" ); document.write( "\n" ); document.write( "Where:\r
\n" ); document.write( "\n" ); document.write( "* t_c = critical t-value (from t-distribution table)
\n" ); document.write( "* s = sample standard deviation
\n" ); document.write( "* n = sample size\r
\n" ); document.write( "\n" ); document.write( "**1. Commute Time**\r
\n" ); document.write( "\n" ); document.write( "* Sample Mean (x̄) = 35.5 minutes
\n" ); document.write( "* Sample Standard Deviation (s) = 7.2 minutes
\n" ); document.write( "* Sample Size (n) = 8
\n" ); document.write( "* Confidence Level = 95%\r
\n" ); document.write( "\n" ); document.write( "* **Degrees of Freedom (df):** df = n - 1 = 8 - 1 = 7
\n" ); document.write( "* **Critical t-value (t_c):** For a 95% confidence level and df = 7, t_c ≈ 2.365 (from a t-distribution table)
\n" ); document.write( "* **Margin of Error (E):** E = 2.365 * (7.2 / √8) ≈ 6.02 minutes
\n" ); document.write( "* **Confidence Interval:** 35.5 ± 6.02 = (29.48, 41.52) minutes\r
\n" ); document.write( "\n" ); document.write( "**Interpretation:** We are 95% confident that the true population mean commute time to work is between 29.48 and 41.52 minutes.\r
\n" ); document.write( "\n" ); document.write( "**2. Microwave Repair Cost**\r
\n" ); document.write( "\n" ); document.write( "* Sample Mean (x̄) = $80.00
\n" ); document.write( "* Sample Standard Deviation (s) = $13.50
\n" ); document.write( "* Sample Size (n) = 13
\n" ); document.write( "* Confidence Level = 95%\r
\n" ); document.write( "\n" ); document.write( "* **Degrees of Freedom (df):** df = n - 1 = 13 - 1 = 12
\n" ); document.write( "* **Critical t-value (t_c):** For a 95% confidence level and df = 12, t_c ≈ 2.179 (from a t-distribution table)
\n" ); document.write( "* **Margin of Error (E):** E = 2.179 * (13.50 / √13) ≈ $8.15
\n" ); document.write( "* **Confidence Interval:** $80.00 ± $8.15 = ($71.85, $88.15)\r
\n" ); document.write( "\n" ); document.write( "**Interpretation:** We are 95% confident that the true population mean repair cost for microwaves is between $71.85 and $88.15.\r
\n" ); document.write( "\n" ); document.write( "**3. Computer Repair Cost**\r
\n" ); document.write( "\n" ); document.write( "* Sample Mean (x̄) = $110.00
\n" ); document.write( "* Sample Standard Deviation (s) = $44.50
\n" ); document.write( "* Sample Size (n) = 7
\n" ); document.write( "* Confidence Level = 95%\r
\n" ); document.write( "\n" ); document.write( "* **Degrees of Freedom (df):** df = n - 1 = 7 - 1 = 6
\n" ); document.write( "* **Critical t-value (t_c):** For a 95% confidence level and df = 6, t_c ≈ 2.447 (from a t-distribution table)
\n" ); document.write( "* **Margin of Error (E):** E = 2.447 * (44.50 / √7) ≈ $41.06
\n" ); document.write( "* **Confidence Interval:** $110.00 ± $41.06 = ($68.94, $151.06)\r
\n" ); document.write( "\n" ); document.write( "**Interpretation:** We are 95% confident that the true population mean repair cost for computers is between $68.94 and $151.06.
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