Politicians are interested in knowing the opinions of their constituents on important issues. One administrative assistant to a senator claims that more than 63% of adult women favor stricter gun laws. A recent telephone survey of 1026 adults by IBR Polls found that 65.9% of adult women favored stricter gun laws. Test the claim that significantly more than 63% of adult women favor stricter gun laws. Use a 5% significance level.
How many standard deviations is the sample proportion above/below the population proportion? Choose the best answer from below.
A) 1.45
B) 0.75
C) 2.12
D) 1.92
To test the claim that significantly more than 63% of adult women favor stricter gun laws, we can use a hypothesis test comparing the sample proportion to the population proportion.
The null hypothesis (H0) is that the population proportion is equal to 63%.
The alternative hypothesis (Ha) is that the population proportion is significantly more than 63%.
In this case, because we are testing if the proportion is significantly more than 63%, we would perform a one-tailed test with the alternative hypothesis being greater than.
Let's calculate the test statistic to determine how many standard deviations the sample proportion is above the population proportion.
First, we calculate the standard error (SE) of the sample proportion:
SE = sqrt((p * (1 - p)) / n)
Where p is the sample proportion (0.659) and n is the sample size (1026).
SE = sqrt((0.659 * (1 - 0.659)) / 1026)
SE ≈ 0.013
Next, we calculate the z-score, which indicates how many standard deviations the sample proportion is above the population proportion:
z = (x - μ) / SE
Where x is the sample proportion (0.659), μ is the population proportion (0.63), and SE is the standard error.
z = (0.659 - 0.63) / 0.013
z ≈ 2.23
Now we can compare the z-score to the critical value at a 5% significance level. Since we are performing a one-tailed test with the alternative hypothesis of greater than, the critical value is approximately 1.645 (corresponding to a 5% significance level).
Since the calculated z-score (2.23) is greater than the critical value (1.645), we can reject the null hypothesis and conclude that there is significant evidence that more than 63% of adult women favor stricter gun laws.
To answer the question, the sample proportion is approximately 2.23 standard deviations above the population proportion. Therefore, the correct answer is:
C) 2.12