In a psychology experiment in which 100 volunteers were asked to read a paragraph about an engineer, 65 assumed that the engineer was male despite the fact that the paragraph did not specify gender (and avoided gendered pronouns such as “he” or “she”). If the null hypothesis here is that there is no gender bias, what is the two-sided p-value associated with this result? Use a normal approximation to solve this.
P=.0001 p=.003 p=.10 p=.05 p=.99
0.05
0.003
.0001
To calculate the two-sided p-value using a normal approximation, we need to follow these steps:
1. Define the null and alternative hypotheses:
- Null hypothesis (H0): There is no gender bias, and the assumption that 50% of the volunteers would assume the engineer as male is correct.
- Alternative hypothesis (Ha): There is a gender bias, and the assumption that 50% of the volunteers would assume the engineer as male is incorrect.
2. Determine the test statistic:
The test statistic for this experiment is the number of volunteers who assumed the engineer was male out of the total 100 volunteers.
3. Calculate the expected value under the null hypothesis:
The expected value, assuming no gender bias, would be half of the total number of volunteers. Therefore, the expected value under H0 is 100 * 0.5 = 50.
4. Calculate the standard deviation under the null hypothesis:
The standard deviation (sigma) is the square root of the expected value times (1 - expected value) = sqrt(50 * (1 - 0.5)) = sqrt(50 * 0.5) = sqrt(25) = 5.
5. Calculate the z-score:
The z-score is calculated by subtracting the expected value from the observed value, and then dividing by the standard deviation: (65 - 50) / 5 = 3.
6. Calculate the p-value:
Since this is a two-sided test, we need to calculate the p-value for both directions of the distribution.
For the positive side, we will use the Z-table or a calculator to find the area to the right of the calculated z-score (3). The probability associated with this area is approximately 0.0013.
For the negative side, we need to find the area to the left of the calculated z-score (-3), which is also approximately 0.0013.
Now, we add these two probabilities together to get the two-sided p-value: 0.0013 + 0.0013 = 0.0026.
Therefore, the two-sided p-value associated with this result is approximately 0.0026, which is closer to p = 0.003.