In a study of the relationship between pet ownership and physical activity in older adults, 575 subjects reported that they owned a pet, while 1907 reported that they did not. Give a 90% confidence interval for the proportion of older adults in this population who are pet owners. (Round your answers to three decimal places.)
Lower limit ____________
Upper limit ____________
To calculate the 90% confidence interval for the proportion of older adults who are pet owners, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
First, let's calculate the sample proportion of pet owners:
Sample Proportion = Number of pet owners / Total number of subjects
= 575 / (575 + 1907)
= 575 / 2482
= 0.2319
Next, we need to find the margin of error. The formula for margin of error is:
Margin of Error = Critical value * Standard deviation
Since we don't know the standard deviation of the population, we will use the standard deviation of the sample proportion, which is:
Standard Deviation = sqrt((Sample Proportion * (1 - Sample Proportion)) / n)
Where n is the total number of subjects.
Standard Deviation = sqrt((0.2319 * (1 - 0.2319)) / 2482)
= sqrt(0.1812 / 2482)
= sqrt(0.000073)
= 0.0085
To find the critical value for a 90% confidence interval, we need to look up the z-score using a standard normal distribution table. For a 90% confidence level, the z-score is approximately 1.645.
Now we can calculate the margin of error:
Margin of Error = Critical value * Standard deviation
= 1.645 * 0.0085
= 0.0139775
Finally, we can calculate the lower and upper limits of the confidence interval:
Lower limit = Sample Proportion - Margin of Error
= 0.2319 - 0.0139775
= 0.2179225
Upper limit = Sample Proportion + Margin of Error
= 0.2319 + 0.0139775
= 0.2458775
Therefore, the 90% confidence interval for the proportion of older adults who are pet owners is:
Lower limit = 0.218
Upper limit = 0.246