2023 Summer
Home
Announcements
Modules
Grades
A random sample of 12 pregnant dogs was selected and produced the following gestation times, in days.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline 63.1 & 65.9 & 63.3 & 58.3 & 63.8 & 67.5 & 63.2 & 59.1 & 63.7 & 61.2 & 60.2 & 63.6 \\
\hline
\end{tabular}
Note: $\sum x^{2}=47,316.07$
a) Find a $95 \%$ confidence interval for the true mean gestation time for pregnant dogs.
b) Interpret the interval in terms of the application.
c) Suppose we were not happy with the margin of error in part (a). What is the minimum sample size required to ensure that our estimate will be within 0.67 days of the true mean gestation time?

Step 1 ：Given a random sample of 12 pregnant dogs with the following gestation times in days: 63.1, 65.9, 63.3, 58.3, 63.8, 67.5, 63.2, 59.1, 63.7, 61.2, 60.2, 63.6.

Step 2 ：We are asked to find a 95% confidence interval for the true mean gestation time for pregnant dogs. The formula for a confidence interval for a mean is given by: \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level (for a 95% confidence level, \(z \approx 1.96\)), \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 3 ：First, we calculate the sample mean, which is the sum of the observations divided by the number of observations. The sample mean is approximately 62.74.

Step 4 ：Next, we calculate the sample standard deviation, which is the square root of the variance. The variance is the sum of the squared deviations from the mean divided by the number of observations minus 1. The sample standard deviation is approximately 2.66.

Step 5 ：Using the z-score for a 95% confidence level, which is approximately 1.96, we can calculate the margin of error, which is approximately 1.51.

Step 6 ：Finally, we calculate the confidence interval by adding and subtracting the margin of error from the sample mean. The 95% confidence interval for the true mean gestation time for pregnant dogs is approximately \((61.24, 64.25)\) days.

Step 7 ：Final Answer: The $95 \%$ confidence interval for the true mean gestation time for pregnant dogs is \(\boxed{(61.24, 64.25)}\) days.