The standard deviation for the low-income group is: 3.183
The standard deviation for the high-income group is 3.101
The correct size of a nickel is 21.21 millimeters. Based on that, the data can be summarized into the following table:
Too Small Too Large Total
Low Income 14 26 40
High Income 26 9 35
Total 40 35 75
find a 90% confidence interval for the proportion of children in the low-income group that drew the nickel too large.
Round all answers to 3 decimal places.
(a) Critical value (positive value only):
(b) Margin of error:
(c) Confidence interval:
_____< p <_____
Use the sample data from the previous problems to find a 98% confidence interval for the mean nickel diameter drawn by the children in the high-income group.
(e) Critical value (positive value only):
(f) Margin of error:
(g) Confidence interval:
_____< μ < ______
To find the 90% confidence interval for the proportion of children in the low-income group that drew the nickel too large, we can follow these steps:
Step 1: Calculate the sample proportion (p̂):
p̂ = Too Large / Total for Low Income Group
= 26 / 40
= 0.65
Step 2: Calculate the standard error (SE):
SE = sqrt((p̂(1 - p̂)) / n)
= sqrt((0.65(1 - 0.65)) / 40)
= sqrt(0.2275 / 40)
= sqrt(0.0056875)
≈ 0.0754
Step 3: Calculate the critical value (positive value only):
Since the confidence level is 90%, we subtract the significance level (1 - 90%) / 2 from 1 and match it to the closest value in the z-table. This gives us a critical value of 1.645.
(a) Critical value (positive value only): 1.645
Step 4: Calculate the margin of error (ME):
ME = Critical value * SE
= 1.645 * 0.0754
≈ 0.124
(b) Margin of error: 0.124
Step 5: Calculate the confidence interval:
Confidence interval = p̂ ± ME
= 0.65 ± 0.124
(c) Confidence interval: 0.526 < p < 0.774
To find the 98% confidence interval for the mean nickel diameter drawn by the children in the high-income group, we can follow these steps:
Step 1: Calculate the sample mean (x̄):
x̄ = Too Large group mean diameter
Step 2: Calculate the standard deviation (s):
s = Standard deviation for the high-income group
= 3.101
Step 3: Calculate the standard error (SE):
SE = s / sqrt(n)
= 3.101 / sqrt(35)
Step 4: Calculate the critical value (positive value only):
Since the confidence level is 98%, we subtract the significance level (1 - 98%) / 2 from 1 and match it to the closest value in the t-table with degrees of freedom (df) equal to (n - 1). This gives us a critical value of t.
(e) Critical value (positive value only): Find t-value in the t-table with df = (n - 1)
Step 5: Calculate the margin of error (ME):
ME = Critical value * SE
(f) Margin of error: Calculate using the critical value and standard error.
Step 6: Calculate the confidence interval:
Confidence interval = x̄ ± ME
(g) Confidence interval: Fill in the calculated values for x̄, ME, and the appropriate critical value in the confidence interval expression.