Question

IQ tests are scaled so that the mean score in a largepopulation should be μ =100. We suspect that the very-low-birth-weight population has mean score less than100. Infants weiging less than 1500 grams at birth are classed as "very low birth weight". Low birth weight carriesmany risks. One study followed 113 male infants with very low birth weight to adulthood. At age 20, the mean IQ score for these men was (x bar=87.6.) Iq scores vary Normally with standard deviation σ=15. Give a 95% confidence interval for the mean IQ score at age 20 for allvery-low-birth-weight males. Use the four-step process for confidence interval.

Answer 1

The 95% confidence interval is
`84.83< \mu < 90.37`

Explanation:

From the question we are told that

The sample size is
`n = 113`

The sample mean is
`\= x = 87.6`

The standard deviation is
`\sigma = 15`

Given that the confidence level is 95% then the level of significance is mathematically represented as

`\alpha = 100 - 95`

`\alpha = 5\%`

`\alpha = 0.05`

Next we obtain the critical value of
`(\alpha )/(2)` from the normal distribution table, the value is

`Z_{( \alpha )/(2) } = 1.96`

Generally the margin of error is mathematically evaluated as

`E = Z_{(\alpha )/(2) } * ( \sigma)/( √(n) )`

=>
`E = 1.96 * ( 15)/( √(113) )`

=>
`E = 2.77`

The 95% confidence interval is mathematically represented as

`\= x - E < \mu < \= x + E`

substituting values

`87.6 - 2.77< \mu < 87.6 + 2.77`

`84.83< \mu < 90.37`