Question

Birth weight and IQ: Infants weighting less than 1500 grams at birth are classed as a very low birth weight. Low birth weight carries many risks. One study followed 116 male infants with very low birth weight to adulthood. At age 20, the mean IQ score for these 116 men was 96.1. IQ scores vary normally with standard deviation σ = 15.

1. (a) Give a 96% CI for the mean IQ score at age 20 for all very low birth weight males. (5pts)

2. (b) Interpret the CI you obtained in part (a). (3pts)

3. (c) To construct a 96% CI for the mean IQ score at age 20 for all very low birth weight males, how large a sample do you need if you want the margin of error to be within 2.5. (5pts)

4. (d) We suspect that the very low birth weight population has mean IQ score less than 100. Does the study described above give good evidence that this is true?

   i. List the null and alternative hypothesis. (3pts) ii. Calculate the test statistic. (3pts)

   iii. Find the p-value for the test. (3pts)

   iv. Draw your conclusion using α = 4% and answer the research question.(3pts)

Answer 1

(a)

Standard error of mean = = 15 / = 1.392715

Since we do know the population standard deviation, we will use z statistic to compute the confidence interval.

z score for 96% confidence interval is 2.054

96% confidence interval is,

96.1 - 2.054 * 1.392715 to 96.1 + 2.054 * 1.392715

93.24 to  98.96

(b)

We can be 95% confident, based on the method used to construct the interval, that the true mean IQ score at age 20 for all very low birth weight males is between 93.24 and 98.96.

(c)

Margin of Error, E = 2.5

Sample size, n =

(d)

i.

ii.

Test statistic, z = (96.1 - 100) / 1.392715 = -2.80

iii.

p-value = p(z < -2.80) =  0.0026

iv.

Since, p-value is less than 0.04 significance level, we reject null hypothesis H0 and conclude that there is strong evidence that true mean IQ score at age 20 for all very low birth weight males is less than 100.