A random survey of 500 households
determines that 20% of the households would do so and that each participating
household expects to recycle about 100 cans monthly with a standard deviation
of 30 cans.
Question
(a) What is the 95%
confidence interval regarding the percentage of households expected to
recycle?
(b) what is the 95% confidence interval regarding the number of cans to be recycled?
is 7.02 correct for A
and I am still stuck on B
To calculate the confidence intervals regarding the percentage of households expected to recycle and the number of cans to be recycled, you can use the formula for calculating confidence intervals based on sample means and proportions.
(a) 95% Confidence Interval for the Percentage of households expected to recycle:
To calculate the confidence interval for the percentage of households expected to recycle, we need to use the sample proportion and the margin of error formula.
Sample Proportion (p̂) = 0.20 (20%) - Since the survey found that 20% of the households recycle.
Margin of Error (E) = Z * sqrt((p̂ * (1 - p̂)) / n) - The margin of error depends on the desired confidence level and the sample size. Z represents the critical value, which corresponds to the desired confidence level (in this case, 95%), and n represents the sample size.
Since the sample size is 500, we can use the Z-distribution for a 95% confidence level, which has a critical value of 1.96 (the value with 2.5% in each tail).
Plugging in the values, the formula becomes:
Margin of Error (E) = 1.96 * sqrt((0.20 * (1 - 0.20)) / 500)
Now, to calculate the confidence interval, we subtract and add the margin of error to the sample proportion:
Confidence Interval = p̂ ± E
Calculating the confidence interval:
Confidence Interval = 0.20 ± (1.96 * sqrt((0.20 * (1 - 0.20)) / 500))
(b) 95% Confidence Interval for the Number of cans to be recycled:
To calculate the confidence interval for the number of cans to be recycled, we use the sample mean and the margin of error formula.
Sample Mean (x̄) = 100 (mean number of cans expected to be recycled)
Sample Standard Deviation (s) = 30 (standard deviation of cans recycled)
Margin of Error (E) = Z * (s / sqrt(n))
Using the same Z-distribution critical value of 1.96 for a 95% confidence level, the formula becomes:
Margin of Error (E) = 1.96 * (30 / sqrt(500))
Confidence Interval = x̄ ± E
Calculating the confidence interval:
Confidence Interval = 100 ± (1.96 * (30 / sqrt(500)))
Now you can substitute the values into the formulas to find the confidence intervals for both the percentage of households expected to recycle and the number of cans to be recycled.