Question

A 99% confidence interval (in inches) for the mean height of a population is 65.67 < μ < 67.13. This result is based on a sample of size 144. If the confidence interval 65.87 < μ < 66.93 is obtained from the same sample data, what is the degree of confidence?

Answer 1

Confidence = 0.94 = 94%

Explanation:

Solution:-

- A 99% confidence interval (in inches) for the mean height of a population is:

65.44 < μ < 66.96

- If the confidence interval is obtained from the same sample data:

65.65 < μ < 66.75

- The sample size, n = 144

- You will first need to find the sample mean (x_bar) and sample standard deviation (s) based on the confidence interval given. The width of the confidence interval is 2E

2E = 66.96-65.44 = 1.52

E = 0.76

- The test statistic error (E) is defined as:

`E = z-critical*(s)/(√(n) )`

Where, Z-critical for 99% confidence = 2.5758

`s = E*(√(n) )/(z-critical) \\\\s = 0.76*(√(144) )/(2.5758)\\\\s=3.54`

- Since,

x_bar - E = 65.44

x_bar = 65.44 - 0.76 = 66.2

- Use the value you found in part a to determine the degree of confidence for the interval 65.65 < μ < 66.75 is based on:

66.2 - E = 65.65

E = 0.55

- The test statistic error (E) is defined as:

`E = z-critical*(s)/(√(n) )`

- Determine the Z-critical value from equation above:

`z-critical = E*(√(n) )/(s) \\\\z-critical = 0.55*(√(144) )/(3.54)\\\\z-critical=1.864`

- The level of confidence for the corresponding Z-critical value would be:

Confidence = P ( - z-critical < Z < z-critical )

Confidence = P ( - 1.864 < Z < 1.864 )

Confidence = 0.94 = 94%