In a survey of 633 males ages 18-64, 395 say they have gone to the dentist in the past year. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. The 90% confidence interval for the population proportion p is?
To calculate confidence intervals for population proportions, we can use the following formula:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
Where:
- Sample Proportion is the proportion from our sample (395/633 in this case).
- Z is the critical value from the standard normal distribution corresponding to the desired level of confidence (90% or 95% in this case).
- Standard Error is the calculated standard deviation of the sample proportion.
First, let's calculate the sample proportion (p̂):
p̂ = (395/633) = 0.6237 (rounded to 4 decimal places)
Then, we need to calculate the standard error:
Standard Error = sqrt( (p̂ * (1 - p̂)) / n )
Where n is the sample size (633 in this case).
Standard Error = sqrt( (0.6237 * (1 - 0.6237)) / 633 )
= 0.0183 (rounded to 4 decimal places)
Now, we need to find the critical value (Z) for the desired level of confidence (90% or 95%). For a 90% confidence level, the Z-value is approximately 1.645, while for a 95% confidence level, the Z-value is approximately 1.96.
Let's calculate the confidence intervals for both levels of confidence:
For a 90% confidence level:
Confidence Interval = 0.6237 ± (1.645 * 0.0183)
= 0.6237 ± 0.0301
= (0.5936, 0.6538)
For a 95% confidence level:
Confidence Interval = 0.6237 ± (1.96 * 0.0183)
= 0.6237 ± 0.0359
= (0.5878, 0.6596)
Interpretation:
For a 90% confidence level, we can say that the true proportion of males ages 18-64 who have gone to the dentist in the past year is estimated to be between 59.36% and 65.38%.
For a 95% confidence level, we can say that the true proportion of males ages 18-64 who have gone to the dentist in the past year is estimated to be between 58.78% and 65.96%.
Comparison of Confidence Interval Widths:
The width of the confidence interval is determined by the critical value (Z) and the standard error. Looking at the two confidence intervals, we can observe that the 95% confidence interval is wider than the 90% confidence interval. This means that the 95% confidence interval is more conservative (or wider) and provides a higher level of confidence that the true population proportion lies within the interval. However, the increased level of confidence comes at the cost of having a wider range of values.