1. An operations manager plans to select 300 female employees from a
group of tenure workers (where height was considered due the nature of
their task). The selected group has an average height of 170 cm and a
sample standard deviation of 25 cm. What is the 95% confidence interval
of all the employees' heights? confidence interval of all the employees'
heights?
SOLUTION:
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Answer:
To calculate the 95% confidence interval for all the employees' heights, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
First, let's calculate the standard error, which represents the variability of the sample mean:
Standard Error = Sample Standard Deviation / √(Sample Size)
Given:
Sample Mean (x̄) = 170 cm
Sample Standard Deviation (s) = 25 cm
Sample Size (n) = 300
Standard Error = 25 / √300
Next, we need to determine the critical value for a 95% confidence level. Since the sample size is large (n > 30), we can use the Z-score table to find the critical value. The critical value for a 95% confidence level is approximately 1.96.
Now we can calculate the confidence interval:
Confidence Interval = 170 ± (1.96 * (25 / √300))
Calculating the value inside the parentheses:
1.96 * (25 / √300) ≈ 1.96 * 1.44 ≈ 2.82
Confidence Interval = 170 ± 2.82
Therefore, the 95% confidence interval for all the employees' heights is approximately (167.18, 172.82) cm. This means that we can be 95% confident that the true average height of all the employees falls within this range.