Is the p-value to the below question 0.4995? Do you reject or fail to reject the null hypothesis?
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.
To determine whether the p-value for the given question is 0.4995 and to decide whether to reject or fail to reject the null hypothesis, we need to conduct a hypothesis test.
Here's how you can do it step by step:
Step 1: State the hypotheses
- Null hypothesis (H0): The proportion of adult women favoring stricter gun laws is equal to or less than 63%.
- Alternative hypothesis (Ha): The proportion of adult women favoring stricter gun laws is greater than 63%.
Step 2: Set the significance level (α)
- The significance level, α, is the threshold that we use to determine if the results are statistically significant. Let's assume it is 0.05 (5%).
Step 3: Compute the test statistic and p-value
To calculate the test statistic, we need the sample proportion, the expected proportion under the null hypothesis, and the standard error.
- The sample proportion is given as 65.9% (0.659).
- The expected proportion under the null hypothesis is 63% (0.63).
The standard error formula for proportions is:
Standard Error = √[(p * (1 - p)) / n], where p is the expected proportion and n is the sample size.
Plugging our values into the formula:
Standard Error = √[(0.63 * (1 - 0.63)) / 1026]
Standard Error ≈ 0.0129
We can then calculate the test statistic using the formula:
Test Statistic = (Sample Proportion - Expected Proportion) / Standard Error
Test Statistic = (0.659 - 0.63) / 0.0129 ≈ 2.25
To find the p-value, we need to compare the test statistic to the appropriate distribution, in this case, a t-distribution or a normal distribution.
Step 4: Determine the p-value and make a decision
To determine the p-value, you would perform a one-sample z-test or a one-sample t-test, depending on the sample size and whether the population standard deviation is known.
Given that we don't have information about the standard deviation of the population, a one-sample t-test is appropriate in this case.
You can then use statistical software or a t-table to find the p-value associated with the test statistic of 2.25. Alternatively, you can use an online calculator or a statistical software package.
Based on our assumptions and calculations, if the p-value is 0.4995, we would compare it to the significance level (α). If the p-value is greater than α (i.e., p-value > 0.05), we would fail to reject the null hypothesis. If the p-value is less than or equal to α (i.e., p-value ≤ 0.05), we would reject the null hypothesis.
In conclusion, without knowledge of the exact p-value, we cannot determine whether to reject or fail to reject the null hypothesis of this hypothesis test.