n a survey of 634 males ages 18-64, 392 say that they have gone to the dentist in the past year

contrast 90% and 95% confidence intervals for the population proportions, interperate the results and compare withsof the confidence intervals

To calculate the confidence intervals for the population proportions, we can use the formula:

Confidence Interval = Sample Proportion ± Margin of Error

To find the margin of error, we need to calculate the standard error using the formula:

Standard Error = √(p̂(1-p̂)/n)

where p̂ is the sample proportion and n is the sample size.

Let's start by calculating the sample proportion for the males who have gone to the dentist in the past year:

Sample Proportion (p̂) = Number of males who have gone to the dentist / Total number of males surveyed
= 392 / 634
= 0.617

Now, let's calculate the standard error for both the 90% and 95% confidence intervals:

For the 90% confidence interval:
Standard Error = √(0.617(1-0.617)/634)
= 0.015

For the 95% confidence interval:
Standard Error = √(0.617(1-0.617)/634)
= 0.015

Now, we can calculate the margin of error for both confidence intervals:

For the 90% confidence interval:
Margin of Error = Z * Standard Error

The Z-value for a 90% confidence interval is approximately 1.645 (which can be found in a Z-table). Therefore,

Margin of Error = 1.645 * 0.015
= 0.025

For the 95% confidence interval:
Margin of Error = Z * Standard Error

The Z-value for a 95% confidence interval is approximately 1.96 (which can be found in a Z-table). Therefore,

Margin of Error = 1.96 * 0.015
= 0.029

Now, let's calculate the confidence intervals:

For the 90% confidence interval:
Lower bound = Sample Proportion - Margin of Error
= 0.617 - 0.025
= 0.592

Upper bound = Sample Proportion + Margin of Error
= 0.617 + 0.025
= 0.642

So the 90% confidence interval is [0.592, 0.642].

For the 95% confidence interval:
Lower bound = Sample Proportion - Margin of Error
= 0.617 - 0.029
= 0.588

Upper bound = Sample Proportion + Margin of Error
= 0.617 + 0.029
= 0.646

So the 95% confidence interval is [0.588, 0.646].

Interpreting the results:
The 90% confidence interval suggests that we can be 90% confident that the true proportion of males who have gone to the dentist in the past year lies between 0.592 and 0.642. Similarly, the 95% confidence interval suggests that we can be 95% confident that the true proportion lies between 0.588 and 0.646.

Comparing the widths of the two confidence intervals:
The width of the 90% confidence interval (0.050) is narrower than the width of the 95% confidence interval (0.058). This means that we are more certain (with a higher level of confidence) about the true proportion when using a 90% confidence interval, as it provides a smaller range of values.

Keep in mind that increasing the confidence level (e.g., from 90% to 95%) results in a wider interval, providing more precision at the cost of a larger range. The choice of the confidence level depends on the desired trade-off between precision and risk of error.