Question

Your co-worker, Jim, randomly draws a sample of 16 of your customers and calculates that 95% of the businesses have sales between $38,000,000 and $68,000,000. He bases his conclusion on a 95% confidence interval using the properties of the Central Limit Theorem. Jim estimated the standard deviation of the population from the standard deviation of the sample, and then used the z-distribution to calculate the confidence interval.

You have an understanding of statistics principles and immediately see an error in his calculations; What is the error and how can you correct it?

Answer 1

(a)

Error = Central Limit Theorem is valid only for large sample. Here Sample Size =n = 16 < 30. Small Sample. So,Central Limit Theorem cannot be applied. Z distribution is not applicable. Only t test is to be used.

(b)

It an be corrected as follows:

Margin of error (E) is given by:

So,

For = 0.05, Z = 1.96

Thus, we get:

So,

For ndf = 16 - 1 = 15 and = 0.05,

From Table, critical values of t = 2.1314

So

Corrected Confidence Interval:

53,000,000 (2.1314 X 7,653,061.2245)

= 53,000,000 16,311,734.6939

= ( 36,688,265.3061 ,69,311,734.6939)

So,

Corrected Confidence Interval:

36,688,265.31 < < 69,311,734.69