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 1490 undergraduate students showed that this percentage is 31%. Can you reject the null hypothesis at a 5% 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 450. 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. observed Z= p-value = Critical value = Hence we can conclude that the percentage of undergraduate students in the U.S. who show almost no gain in learning in their first two years of college is currently 45%

showing work would be helpful! thank you

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis: p = 0.45
Alternative Hypothesis: p < 0.45

Rejection Region
This is left tailed test, for α = 0.1
Critical value of z is -1.28.
Hence reject H0 if z < -1.28

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.31 - 0.45)/sqrt(0.45*(1-0.45)/1490)
z = -10.86

P-value Approach
P-value = 0
As P-value < 0.1, reject the null hypothesis.

observed z = -10.86

p-value = 0.0000

critical valeu = -1.28

Less than 45%