Question

Researchers studying the human voice find that 22 of the 50 opera singers in a sample can reach volumes

over 100 decibels. Carry out a hypothesis test with alpha= 0.1 to determine the proportion of professional opera

singers who can reach over 100 decibels is less than 0.5.

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.5
Alternative Hypothesis, Ha: p < 0.5

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 = (0.44 - 0.5)/sqrt(0.5*(1-0.5)/50)
z = -0.85

P-value Approach
P-value = 0.1977
As P-value >= 0.1, fail to reject null hypothesis.