In a survey of 621 males ages 18-64 , 396 say they have gone to the dentist in the past year.
Construct 90% and 95% confidence intervals for the proportion. Interpret the results and compare the widths of the confidence intervals.
Please show step by step. Thank you!
Use confidence interval formulas for proportions.
CI95 = p + or - (1.96)(√pq/n)
...where √ = square root, p = x/n, q = 1 - p, and n = sample size.
CI90 = p + or - (1.645)(√pq/n)
Hint: x = 396, n = 621
Convert all fractions to decimals to work the formulas.
I hope this will help get you started.
To construct confidence intervals for the proportion, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
Step 1: Calculate the sample proportion (p̂):
The sample proportion is the proportion of males who have gone to the dentist in the past year, which is given as 396 out of 621.
p̂ = 396 / 621
Step 2: Calculate the standard error (SE):
The standard error measures the variability in the sample proportion. It is calculated using the formula:
SE = sqrt(p̂ * (1 - p̂) / n)
Where n is the sample size, which is 621 in this case.
SE = sqrt(p̂ * (1 - p̂) / n)
SE = sqrt((396 / 621) * (1 - 396 / 621) / 621)
Step 3: Calculate the margin of error (ME):
The margin of error determines the range around the sample proportion within which the population proportion is likely to fall. It is calculated using the formula:
ME = Critical Value * SE
The critical value depends on the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645, and for a 95% confidence level, the critical value is approximately 1.96.
For the 90% confidence interval:
ME = 1.645 * SE
For the 95% confidence interval:
ME = 1.96 * SE
Step 4: Calculate the confidence intervals:
Using the sample proportion, standard error, and margin of error, we can calculate the confidence intervals.
For the 90% confidence interval:
Lower Limit = p̂ - ME
Upper Limit = p̂ + ME
For the 95% confidence interval:
Lower Limit = p̂ - ME
Upper Limit = p̂ + ME
Step 5: Interpret the results and compare the widths of the confidence intervals:
Interpretation involves stating the range within which we are confident that the true population proportion lies. For example:
For the 90% confidence interval, we can say, "We are 90% confident that the true proportion of males who have gone to the dentist in the past year lies between [Lower Limit, Upper Limit]."
For the 95% confidence interval, we can say, "We are 95% confident that the true proportion of males who have gone to the dentist in the past year lies between [Lower Limit, Upper Limit]."
To compare the widths of the confidence intervals, we can look at the margin of error. A wider confidence interval indicates a larger margin of error, which means less precision in estimating the true population proportion. Therefore, the 95% confidence interval will generally be wider than the 90% confidence interval.