A time study od a certain service task found a average cycle time of 15 minutes, with a standard deviation of 5 minutes. These figures were based on a sample of 100 measurements. Is the sample large enough that we are 95% confident that standard time is within 5% of its true value?
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To determine if the sample size is large enough to be 95% confident that the standard time is within 5% of its true value, we need to perform a hypothesis test.
Here's how you can do it step by step:
Step 1: Define the hypothesis:
- Null Hypothesis (H₀): The true standard time is not within 5% of the average cycle time.
- Alternative Hypothesis (H₁): The true standard time is within 5% of the average cycle time.
Step 2: Determine the significance level:
In this case, the confidence level is 95%, which corresponds to a significance level (α) of 0.05.
Step 3: Calculate the standard error:
The standard error (SE) can be calculated using the formula: SE = standard deviation / √sample size.
SE = 5 minutes / √100 = 0.5 minutes.
Step 4: Calculate the margin of error:
The margin of error (ME) can be calculated by multiplying the standard error by the critical value.
For a 95% confidence level, the critical value can be obtained from the standard normal distribution table, which is approximately 1.96.
ME = 1.96 * 0.5 minutes = 0.98 minutes.
Step 5: Calculate the confidence interval:
The confidence interval can be calculated by subtracting the margin of error from the sample mean and adding it to the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
= 15 minutes ± 0.98 minutes
= (14.02 minutes, 15.98 minutes)
Step 6: Conclusion:
Since the confidence interval doesn't include the desired range of ±5% (13.5 minutes to 16.5 minutes), we can conclude that the sample size of 100 is not large enough to be 95% confident that the standard time is within 5% of its true value.
In order to increase the precision and narrow down the confidence interval, a larger sample size would be needed.