Let's break down the problem step-by-step.
1. Calculate the total number of dogs recorded:
The veterinarian's office recorded the following number of dogs on each day:
- Monday: 7 dogs
- Tuesday: 4 dogs
- Wednesday: 5 dogs
- Thursday: 5 dogs
- Friday: 2 dogs
Adding these values together:
[tex]\[
7 + 4 + 5 + 5 + 2 = 23 \text{ dogs}
\][/tex]
2. Determine the sample proportion:
The total number of patients in a week is given as 50.
The sample proportion of dogs is calculated by dividing the total number of dogs by the total number of patients:
[tex]\[
\text{Sample proportion} = \frac{23}{50} = 0.46 \text{ or } 46\%
\][/tex]
3. Standard error for the proportion:
The standard error (SE) of the sample proportion can be calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
where [tex]\(\hat{p}\)[/tex] is the sample proportion, and [tex]\(n\)[/tex] is the sample size.
Here, [tex]\(\hat{p} = 0.46\)[/tex] and [tex]\(n = 50\)[/tex]:
[tex]\[
SE = \sqrt{\frac{0.46 \times (1 - 0.46)}{50}} \approx 0.0705
\][/tex]
4. Calculate the 90% confidence interval:
For a 90% confidence interval, the Z score is approximately 1.645.
The confidence interval is calculated as:
[tex]\[
\text{CI}_{90} = \hat{p} \pm Z \times SE
\][/tex]
Substituting the values, we get:
[tex]\[
\text{CI}_{90} = 0.46 \pm 1.645 \times 0.0705
\][/tex]
This results in:
[tex]\[
\text{CI}_{90} = (0.344, 0.576)
\][/tex]
5. Calculate the 95% confidence interval:
For a 95% confidence interval, the Z score is approximately 1.96.
The confidence interval is calculated as:
[tex]\[
\text{CI}_{95} = \hat{p} \pm Z \times SE
\][/tex]
Substituting the values, we get:
[tex]\[
\text{CI}_{95} = 0.46 \pm 1.96 \times 0.0705
\][/tex]
This results in:
[tex]\[
\text{CI}_{95} = (0.322, 0.598)
\][/tex]
To summarize, based on the given data:
\begin{tabular}{|c|c|}
\hline
Percentage of patients that were dogs & 46\% \\
\hline
[tex]$90 \%$[/tex] confidence interval & (0.344, 0.576) \\
\hline
[tex]$95 \%$[/tex] confidence interval & (0.322, 0.598) \\
\hline
\end{tabular}
