High School

A sample of n=49 was selected. The sample showed a mean of 77 with a standard deviation of 38.64. For a 99.7 percent confidence interval for μ, the value of the multiplier is __________?

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.

  1. 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).
  2. 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].
  3. 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.