Pencil and Paper Exercise 3

Confidence Intervals, Significance Tests (z test)

1. Researchers plan a study of reading ability in eight-year olds.
They run a small pilot study in order to check the variability of the
test scores.  In the pilot study, the standard deviation of the sample
is s=12 points, and therefore they assume initially that the standard
deviation in the population is the same (sigma = 12 points).  The
sample mean is 26 points.

a. The budget allowed 100 pupils to be chosen for the test.  Calculate
   the 95% confidence interval for the population mean.

b. Determine the minimal number of pupils who need to be involved
   in the study in order to determine the population mean with
   an error margin of 5 points or less.

2. We know from transcripts of children's speech that they learn on
the average muu = 6.9 new words per week during the fourth year.  The
standard deviation is sigma = 2.7 You discover new transcripts where
the age of the child has unfortunately not been recorded.  They show
that the child is using an average of 8.2 new words per week.

c. We wish to test the hypothesis that this child is in its fourth 
   year. Formulate H_0 and H_a.
d. Calculate the test statistic z.

e. Determine the p-value.  Wat is your conclusion about the child's
   age (assuming it is normal developmentally and linguistically)?
f. Is the z-score significant at the 5%-level (alfa = 0.05)?