Question

In an August 2012 Gallup survey of 1,012 randomly selected U.S. adults (age 18 and over), 536 said that they were dissatisfied with the quality of education students receive in kindergarten through grade 12.

(a) Test, at 5% significance level, if this sample provides evidence that the proportion of Americans who are dissatisfied with education in kindergarten through grade 12 differs significantly from 50%. (Be sure to state all the 5 steps involved in a hypothesis testing, Hypothesis, Observed statistic, p-value, decision and conclusion in contest of the problem) Hint: Use Statkey to get your p-value.

(b) Is the Test Significant? Why or Why not.

Answer 1

Solution: (a) For the given question we construct our null and alternative hypotheses as:

H0: p = 0.50 vs Ha: p 0.50 [Since we ae to test if the proportion of dissatisfied Americans is significantly different from 50%. So, it is a two tailed test.]

[Here p= unknown population proportion of the dissatisfied Americans with education in kindergarten through grade 12]

The test statistic for this is Z = (p_hat - p0)/sqrt(p0*(1-p0)/n) ~ N(0,1), under H0.

here p0 = hypothetical value of the unknown population proportion, n= sample size, p_hat = sample proportion.

We reject H0 if |Z(observed)| > tau(alpha/2) where tau(alpha/2) is the upper alpha/2 point of a standard normal distribution. Or if p-value is less than the level of significance. p-value being the probability of finding a result more extreme than the observd test statistic, assuming the null hypothesis is true.

Here n= 1012, p_hat = 536/1012, p0=0.5. Also, the observed test statistic is Z(observed) = 1.886084
and tau(alpha/2) =   1.959964.

The obtained p-value is = 0.0592

So, we see that |Z(observed)| < tau(alpha/2), also, p-value > 0.05. Hence we fail to reject H0 and conclude at a 5% level of significance on the basis of the given sample measures that there is not enough evidence to say that the proportion of issatisfied Americans is significantly different from 50%.

(b) The test is not significant, since the obtained p-vaue is not less than the level of significance, alpha.