Question

According to a book published in 2011, 45% of the undergraduate students in the United States show almost no gain in learning in their first two years of college (Richard Arum et al., Academically Adrift, University of Chicago Press, Chicago, 2011). A recent sample of 1530 undergraduate students showed that this percentage is 32%. Can you reject the null hypothesis at a 10% significance level in favor of the alternative that the percentage of undergraduate students in the United States who show almost no gain in learning in their first two years of college is currently lower than 45%. Use both the p-value and the critical-value approaches.

Round your answers for the observed value of z and the critical value of z to two decimal places, and the p-value to four decimal places.

z_observed =

p-value =
Critical value =

Answer 1

Null hypothesis, 0.45

Alternative Hypothesis, 0.45

pcap = 0.32

Test statistic,

z = (0.32 - 0.45)/sqrt(0.45*0.55/1530)
z = -10.22

this is left tailed test,
p-value = P(z < -10.22)
p-value = 0.000
As p-value < 0.1, reject H0

Here the significance level, 0.1. This is right tailed test; hence rejection region lies to the right. -1.28 i.e. P(z < -1.28) = 0.05

Reject H0 if test statistic, z < -1.28

Reject H0