Question

Scores on a standardized vocabulary test for measuring the vocabulary skills of six-year-old children form a normal distribution with = 500 and standard deviation = 60. A developmental psychologist hypothesizes that children's vocabulary skills can be increased by having their parents read out loud during pregnancy. You – Dr. Grisham - are one of several researchers to test this hypothesis by selecting a random sample of pregnant women and having them read out loud during their pregnancies. When you measure their children’s' vocabulary skills six years later the scores are as follows: 475, 455, 448, 515, 522, 485, 535, 490, 465, 555, 585, 595, 495, 585, 515, 500 a.) Given this sample size and variability, how much sampling error should we expect when estimating a population mean from this sample mean? b.) Represent the null and alternative hypotheses in symbols c.) locate your critical region (using α = .05 for a two-tailed test). d.) Based on this sample, what can you conclude about whether reading aloud during pregnancy has an effect on vocabulary skills?

Answer 1

a.)

The sampling error or margin of error is obtained using the following formula,

where the z critical value is obtained from the standard normal distribution table for significance level = 0.05 for two-tailed.

Now,

b.)

The Null and Alternative Hypotheses are,

where is the mean vocabulary score

c.)

Since the population standard deviation is known, the z distribution is used to test the hypothesis.

The critical value for the z is obtained from the z distribution table for significance level = 0.05 and a two-tailed hypothesis.

Hence the rejection region is,

such that reject the null hypothesis if the obtained z statistic is greater than the critical value.

d)

The z statistic is obtained using the formula,

Conclusion:

Since the z statistic is less than 1.96 at a 5% significance level for the two-tailed hypothesis, the null hypothesis is not rejected.

Hence we can conclude that reading aloud during pregnancy has no effect on vocabulary skills