We would like to test the hypothesis that p = 0.5 versus the alternative that p

Question

We would like to test the hypothesis that p = 0.5 versus the alternative that p < 0.5 using data from a random sample. We calculate the standard score to be -1.8. The P-value would then be
A) 0.9641.
B) 0.0359.
C) 0.5000.
D) 0.6915.
E) 0.3085.

Quizplus · Accepted Answer

The standard score (also known as z-score) is given by:
z=  (p_hat - p) / sqrt(p*(1-p) / n)
where p_hat is the sample proportion, p is the hypothesized proportion, n is the sample size, and sqrt is the square root.

Here, we have z = -1.8, so the test statistic is  z=-1.8 = (p_hat - 0.5) / sqrt(0.5*0.5 / n)

Solving for p_hat, we get: 
p_hat = 0.5 + z * sqrt(0.5*0.5 / n) = 0.5 - 1.8 * sqrt(0.25/n)

To find the P-value, we need to find the probability that a standard normal random variable Z is less than or equal to -1.8. From a standard normal distribution table, we find that this probability is approximately 0.0359.

Since this P-value is less than the conventional significance level of 0.05, we reject the null hypothesis at the 5% level of significance and conclude that there is sufficient evidence to support the alternative hypothesis that p < 0.5.

We Would Like to Test the Hypothesis That P =