Answer :
To find the value of the multiplier for a 99.7% confidence interval, we are dealing with a concept in statistics known as the 'confidence level' in the context of a normal distribution.
Understanding Confidence Level:
- The confidence level tells us how certain we are that the true population parameter (in this case, the mean [tex]\mu[/tex]) lies within the confidence interval.
- A 99.7% confidence interval corresponds to approximately 3 standard deviations from the mean (based on the empirical rule, also known as the 68-95-99.7 rule).
Z-Score for 99.7% Confidence Interval:
- When the sample size is large, the sampling distribution of the sample mean is approximately normal.
- For a 99.7% confidence interval, the multiplier is known as the z-score, which corresponds to a very specific point on the normal distribution.
- For a 99.7% confidence interval, that z-score is approximately [tex]z = 3[/tex].
Calculation Process:
- The standard deviation of the sample is given as 38.64, and you have a sample mean of 77, but to find just the multiplier, you don't need these. The value of the multiplier depends on the confidence level and the population distribution.
- Thus, the answer is the z-score multiplier for a 99.7% confidence interval, which is approximately 3.
In conclusion, for this problem, the multiplier (z-value) for a 99.7% confidence interval is approximately [tex]z = 3[/tex]. This means the interval will extend about 3 standard deviations from the sample mean in a normal distribution.
