A purchasing agent at the Kelly Bread Company wants to estimate the mean daily
usage of rye flour. She takes a sample for 50 straight days and finds that the sample
mean is 180 pounds with a sample standard deviation of 38.5 pounds.
a)
State the 90% confidence interval for the mean.
(Round to hundredths)
b)
State the margin of error
(Round to tenths)
To find the confidence interval and margin of error, we can use the formula:
Confidence Interval = sample mean ± margin of error
The margin of error can be calculated using the formula:
Margin of Error = (critical value) x (standard deviation / √sample size)
a) To calculate the 90% confidence interval for the mean, we need to determine the critical value from the t-distribution table for a 90% confidence level with 49 degrees of freedom. Since the sample size is 50 and we have a sample standard deviation, we can use the t-distribution. The critical value for a 90% confidence level and 49 degrees of freedom is approximately 1.676.
Sample size = 50
Sample mean = 180 pounds
Sample standard deviation = 38.5 pounds
Critical value (from t-distribution table) = 1.676
Margin of Error = (1.676) x (38.5 / √50)
Now we can calculate the confidence interval:
Confidence Interval = 180 ± (1.676) x (38.5 / √50)
b) To find the margin of error, we substitute the calculated critical value and sample standard deviation into the margin of error formula:
Margin of Error = (1.676) x (38.5 / √50)
Now we can round the values to the requested decimal places:
a) 90% Confidence Interval = 180 ± (1.676) x (38.5 / √50) = 180 ± 13.19 = (166.81, 193.19)
b) Margin of Error = (1.676) x (38.5 / √50) ≈ 13.2 pounds
To calculate the 90% confidence interval for the mean daily usage of rye flour, we can use the formula:
Confidence Interval = Sample Mean ± (Z * (Sample Standard Deviation / √Sample Size))
a) Now, let's calculate the confidence interval:
Sample Mean = 180 pounds
Sample Standard Deviation = 38.5 pounds
Sample Size = 50 days
Z-value for a 90% confidence level is 1.645 (from the standard normal distribution table).
Confidence Interval = 180 ± (1.645 * (38.5 / √50))
Calculating the expression inside the parentheses first:
= 1.645 * (38.5 / 7.07107) [√50 ≈ 7.07107]
= 1.645 * 5.45354
= 8.9725
Now, calculating the confidence interval:
= 180 ± 8.9725
Therefore, the 90% confidence interval for the mean daily usage of rye flour is approximately:
Confidence Interval = (171.03, 188.97) pounds
b) The margin of error can be found by:
Margin of Error = Z * (Sample Standard Deviation / √Sample Size)
Using the same values from above:
Margin of Error = 1.645 * (38.5 / √50)
= 1.645 * 5.45354
= 8.9725
Therefore, the margin of error is approximately 8.9725 pounds (rounding to tenths).
Confidence interval = mean ± margin of error
CI90 = mean ± 1.645 (sd/√n)
With your data:
CI90 = 180 ± 1.645 (38.5/√50)
I'll let you take it from here.