One of the questions on a survey of 1,000 adults asked if today's children will be better off than their parents (Rasmussen Reports website, October 26, 2012). Representative data are shown in the WEBfile named ChildOutlook. A response of Yes indicates that the adult surveyed did think today's children will be better off than their parents. A response of No indicates that the adult surveyed did not think today's children will be better off than their parents. A response of Not Sure was given by 23% of the adults
a. What is the point estimate of the proportion of the population of adults who do think that today's children will be better off than their parents (to 2 decimals)?
b. At 95% confidence, what is the margin of error (to 4 decimals)?
c. What is the 95% confidence interval for the proportion of adults who do think that today's children will be better off than their parents (to 4 decimals)?
d. What is the 95% confidence interval for the proportion of adults who do not think that today's children will be better off than their parents (to 4 decimals)?
a. The point estimate of the proportion of the population of adults who do think that today's children will be better off than their parents is 0.77.
b. The margin of error at 95% confidence is 0.0246.
c. The 95% confidence interval for the proportion of adults who do think that today's children will be better off than their parents is (0.7454, 0.7946).
d. The 95% confidence interval for the proportion of adults who do not think that today's children will be better off than their parents is (0.2054, 0.2446).
To answer these questions, we need to use statistical calculations based on the survey data. Let's go step by step to find the required estimates:
a. The point estimate of the proportion of adults who think today's children will be better off than their parents can be calculated by dividing the number of 'Yes' responses by the total number of respondents. From the given information, we know that the survey had 1,000 adults and 23% responded with 'Not Sure'. Therefore, the proportion of 'Yes' responses would be (1 - 0.23)%:
Proportion of Yes responses = (1 - 0.23)% = 0.77
So, the point estimate is 0.77.
b. To find the margin of error, we need to calculate the standard error. The formula to calculate the standard error for a proportion is:
Standard Error = sqrt((p * (1 - p)) / n)
where p is the proportion from the point estimate and n is the sample size. In this case, p = 0.77 and n = 1,000.
Standard Error = sqrt((0.77 * (1 - 0.77)) / 1,000)
Using a calculator, the standard error is approximately 0.0139.
The margin of error (ME) at a 95% confidence level can be calculated using the formula:
ME = z * SE
where z is the z-score corresponding to the desired confidence level. For a 95% confidence level, the z-score is 1.96.
ME = 1.96 * 0.0139
Using a calculator, the margin of error is approximately 0.0272 (rounded to 4 decimals).
c. The 95% confidence interval for the proportion of adults who think today's children will be better off than their parents can be calculated by subtracting and adding the margin of error (ME) from the point estimate (p).
Confidence Interval = (p - ME, p + ME)
Substituting the values, the confidence interval is:
CI = (0.77 - 0.0272, 0.77 + 0.0272)
Simplifying, the 95% confidence interval is approximately (0.7428, 0.7972) (rounded to 4 decimals).
d. Similarly, we can calculate the 95% confidence interval for the proportion of adults who do not think that today's children will be better off than their parents.
Since the survey did not provide the proportion of 'No' responses, we can use the complement of the proportion 'Yes' as an estimate. The complement of 0.77 is 1 - 0.77 = 0.23.
So, the 95% confidence interval for the proportion of adults who do not think that today's children will be better off than their parents would be (0.23 - ME, 0.23 + ME).
Substituting the margin of error (ME) value, the confidence interval is:
CI = (0.23 - 0.0272, 0.23 + 0.0272)
Simplifying, the 95% confidence interval is approximately (0.2028, 0.2572) (rounded to 4 decimals).
To calculate the point estimate, margin of error, and confidence intervals for this survey data, we can use the following formulas:
Point Estimate: (Number of Yes responses) / (Total number of responses)
Margin of Error: Critical value * Standard Error
Confidence Interval: Point Estimate ± Margin of Error
First, let's gather the data:
Number of Yes responses (adults who think children will be better off): 1,000 - (23% of 1,000)
Total number of responses (sample size): 1,000
a. Point Estimate:
Number of Yes responses = 1,000 - (23% of 1,000) = 1,000 - (0.23 * 1,000) = 1,000 - 230 = 770
Point Estimate = 770 / 1,000 = 0.77 (rounded to 2 decimals)
b. Margin of Error:
We need to calculate the critical value and the standard error first.
Critical value:
Since the sample size is large (more than 30), we can use the Z-table to find the critical value for a 95% confidence level. The critical value for a 95% confidence level is 1.96.
Standard Error:
Standard Error = sqrt((Point Estimate * (1 - Point Estimate)) / Sample Size)
Standard Error = sqrt((0.77 * 0.23) / 1,000) ≈ 0.0157 (rounded to 4 decimals)
Margin of Error:
Margin of Error = Critical value * Standard Error
Margin of Error = 1.96 * 0.0157 ≈ 0.0307 (rounded to 4 decimals)
c. Confidence Interval for Yes responses:
Confidence Interval = Point Estimate ± Margin of Error
Confidence Interval = 0.77 ± 0.0307 ≈ (0.7393, 0.8007) (rounded to 4 decimals)
d. Confidence Interval for No responses:
We know that the sum of Yes and No responses is equal to the total sample size.
Number of No responses = Total sample size - Number of Yes responses = 1,000 - 770 = 230
Proportion of No responses = 230 / 1,000 = 0.23
Using the same formula as for the Confidence Interval for Yes responses, we can calculate the Confidence Interval for No responses:
Confidence Interval = Proportion of No responses ± Margin of Error
Confidence Interval = 0.23 ± 0.0307 ≈ (0.1993, 0.2607) (rounded to 4 decimals)