Question

a computer company that recently developed a new software product wanted to estimate the mean time taken to learn how to use this software by people who are somewhat familiar with computers. a random sample of 12 such persons was selected. the following data give the times taken (in hours) by the persons to learn how to use this software. 1.75 2.35 2.50 1.80 1.60 2.75 2.15 2.15 1.60 2.00 3.05 2.50 construct a 95% confidence interval for the population mean. assume that the times taken by all persons who are somewhat familiar with computers to learn how to use this software are approximately normally distributed. round your answers to two decimal places. lower bound: upper bound:

Explanation:

Step1: Calculate the sample mean ($\bar{x}$)

Sum all the data values: \(1.75 + 2.35+2.50 + 1.80+1.60 + 2.75+2.15+2.15+1.60+2.00+3.05+2.50=26.2\)
Sample size \(n = 12\)
\(\bar{x}=\frac{26.2}{12}\approx2.18\)

Step2: Calculate the sample standard deviation ($s$)

Use the formula \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
\(\sum_{i=1}^{12}(x_{i}-\bar{x})^{2}=(1.75 - 2.18)^{2}+(2.35 - 2.18)^{2}+(2.50 - 2.18)^{2}+(1.80 - 2.18)^{2}+(1.60 - 2.18)^{2}+(2.75 - 2.18)^{2}+(2.15 - 2.18)^{2}+(2.15 - 2.18)^{2}+(1.60 - 2.18)^{2}+(2.00 - 2.18)^{2}+(3.05 - 2.18)^{2}+(2.50 - 2.18)^{2}\)
\(=(- 0.43)^{2}+(0.17)^{2}+(0.32)^{2}+(-0.38)^{2}+(-0.58)^{2}+(0.57)^{2}+(-0.03)^{2}+(-0.03)^{2}+(-0.58)^{2}+(-0.18)^{2}+(0.87)^{2}+(0.32)^{2}\)
\(=0.1849+0.0289+0.1024+0.1444+0.3364+0.3249+0.0009+0.0009+0.3364+0.0324+0.7569+0.1024 = 2.351\)
\(s=\sqrt{\frac{2.351}{11}}\approx0.46\)

Step3: Find the critical value ($t_{\alpha/2}$)

Confidence level \(C = 0.95\), so \(\alpha=1 - C=0.05\) and \(\frac{\alpha}{2}=0.025\)
Degrees of freedom \(df=n - 1=11\)
From the \(t\)-distribution table, \(t_{0.025,11}=2.201\)

Step4: Calculate the margin of error ($E$)

\(E=t_{\alpha/2}\frac{s}{\sqrt{n}}=2.201\times\frac{0.46}{\sqrt{12}}\approx2.201\times0.133\approx0.29\)

Step5: Calculate the confidence interval

Lower bound: \(\bar{x}-E=2.18 - 0.29 = 1.89\)
Upper bound: \(\bar{x}+E=2.18+0.29 = 2.47\)

Answer:

Lower bound: \(1.89\)
Upper bound: \(2.47\)