Question

select the correct answer from each drop - down menu. a veterinarians office recorded one particular week that they had 50 patients. the table shows the recorded number of dogs: monday tuesday wednesday thursday friday 7 4 5 5 2. use the given data to complete the sample proportion and confidence intervals for this situation. percentage of patients that were dogs 90% confidence interval 95% confidence interval

Explanation:

Step1: Calculate total number of dog - patients

$7 + 4+5 + 5+2=23$

Step2: Calculate sample proportion of dog - patients

The sample proportion $\hat{p}=\frac{23}{50}=0.46$ or $46\%$

Step3: Calculate z - scores for confidence intervals

For a 90% confidence interval, the z - score $z_{\alpha/2}=1.645$ (from standard normal distribution table). For a 95% confidence interval, the z - score $z_{\alpha/2}=1.96$.

Step4: Calculate margin of error for 90% confidence interval

$E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.645\sqrt{\frac{0.46\times(1 - 0.46)}{50}}\approx1.645\sqrt{\frac{0.46\times0.54}{50}}\approx1.645\sqrt{\frac{0.2484}{50}}\approx1.645\times0.0705\approx0.116$
The 90% confidence interval is $\hat{p}-E

Step5: Calculate margin of error for 95% confidence interval

$E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.96\sqrt{\frac{0.46\times(1 - 0.46)}{50}}\approx1.96\sqrt{\frac{0.46\times0.54}{50}}\approx1.96\sqrt{\frac{0.2484}{50}}\approx1.96\times0.0705\approx0.138$
The 95% confidence interval is $\hat{p}-E

Answer:

Percentage of patients that were dogs: $46\%$
90% confidence interval: $(34.4\%,57.6\%)$
95% confidence interval: $(32.2\%,59.8\%)$