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A veterinarian's office recorded one particular week that they had 50 patients. The table shows the recorded number of dogs.
\begin{tabular}{|c|c|c|c|c|}
\hline Monday & Tuesday & Wednesday & Thursday & Friday \\
\hline 7 & 4 & 5 & 5 & 2 \\
\hline
\end{tabular}

Use the given data to complete the sample proportion and confidence intervals for this situation.
\begin{tabular}{|c|c|}
\hline Percentage of patients that were dogs & [tex]$\checkmark$[/tex] \\
\hline [tex]$90\%$[/tex] confidence interval & [tex]$\checkmark$[/tex] \\
\hline [tex]$95\%$[/tex] confidence interval & [tex]$\checkmark$[/tex] \\
\hline
\end{tabular}



Answer :

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}