Finding p-value

Question

A newspaper is conducting a statewide survey concerning the race for governor. The newspaper will take a simple random sample of n registered voters and determine X = the number of voters that will vote for the Democratic candidate. Is there evidence that a clear majority of the population will vote for the Democratic candidate? To answer this, they will test the hypotheses H ₀: p = 0.50 versus H _a: p> 0.50. 

 

If n = 120 and X = 64, what is the P-value for this hypothesis test? Make a sketch to approximate the p-value, without consulting R or a z-table.

Option a) p-value < 0.025

Option b) 0.025 < p-value < 0.05

Option c) 0.05 < p-value < 0.5

Option d) p-value > 0.5

Answer 1

To calculate the p-value for this hypothesis test, we can use the normal distribution approximation since the sample size is sufficiently large (n = 120).

First, calculate the sample proportion:

p-hat = X / n = 64 / 120 = 0.5333

Now, calculate the standard error (SE) of the sample proportion:

SE = sqrt((p-hat * (1 - p-hat)) / n)

SE = sqrt((0.5333 * (1 - 0.5333)) / 120)

SE ≈ 0.0737

Next, calculate the z-score:

z = (p-hat - p) / SE

z = (0.5333 - 0.50) / 0.0737 ≈ 0.4512

Now, we can find the p-value associated with this z-score. We are testing H0: p = 0.50 versus Ha: p > 0.50, so we're interested in the right tail of the distribution.

From a standard normal distribution table or calculator, you can find that the area to the right of z = 0.4512 is approximately 0.3257.

Since we're testing whether a clear majority will vote for the Democratic candidate, the p-value represents the probability that we observe a proportion as extreme as 0.5333 or more in favor of the Democratic candidate. Since the p-value is approximately 0.3257, it falls in the range 0.05 < p-value < 0.5.

So, the correct option is:

Option c) 0.05 < p-value < 0.5