A researcher predicts that watching a film on institutionalization will change students’ attitudes about chronically mentally ill patients. The researcher randomly selects a class of 36 students, shows them the film, and gives them a questionnaire about their attitudes. The mean score on the questionnaire for these 36 students is 70. The score for a similar class of students who did not see the film is 75. The standard deviation is 12. Using the five steps of hypothesis testing and the 5% significance level (i.e. alpha = .05), does showing the film change students’ attitudes towards the chronically mentally ill?
To determine if showing the film changes students' attitudes towards the chronically mentally ill, we can conduct a hypothesis test using the five steps of hypothesis testing.
Step 1: State the hypotheses
- Null hypothesis (H₀): The film does not change students' attitudes towards the chronically mentally ill. μ₁ = μ₂
- Alternative hypothesis (H₁): The film changes students' attitudes towards the chronically mentally ill. μ₁ ≠ μ₂
Here, μ₁ represents the population mean score for students who watched the film, and μ₂ represents the population mean score for students who did not watch the film.
Step 2: Set the significance level (α)
In this case, the significance level is given as 5%, or α = 0.05.
Step 3: Collect and analyze the data
The sample mean scores are provided as follows:
- Mean score for the group that watched the film (sample mean) = 70
- Mean score for the group that did not watch the film = 75
The population standard deviation is also given as 12.
Step 4: Calculate the test statistic
We need to calculate the test statistic, which is the z-score, to compare it with the critical value(s).
Given that we have two independent samples with known standard deviation, we can use the formula for the z-score:
z = (x̄₁ - x̄₂) / sqrt((σ₁²/n₁) + (σ₂²/n₂))
where x̄₁ and x̄₂ are the sample means, σ₁ and σ₂ are the population standard deviations, and n₁ and n₂ are the sample sizes.
In this case:
x̄₁ = 70, x̄₂ = 75, σ₁ = 12, σ₂ = 12, n₁ = 36, n₂ = 36
Calculating the z-score:
z = (70 - 75) / sqrt((12²/36) + (12²/36))
Step 5: Determine the critical value(s) and make a decision
We need to compare the calculated value of the test statistic (z-score) with the critical value(s) to determine if we can reject the null hypothesis.
Since our alternative hypothesis is two-tailed (μ₁ ≠ μ₂), we will need to find the critical values for the two tails of the distribution.
Using a significance level (α) of 0.05, we divide it by 2 to get 0.025 for each tail (0.025 on the left and 0.025 on the right). We can use a standard normal distribution table or statistical software to find the critical values associated with 0.025.
For a two-tailed test, the absolute value of the calculated z-score should be greater than the critical value(s) to reject the null hypothesis.
If the calculated z-score is less than the negative critical value or greater than the positive critical value, we can reject the null hypothesis. Otherwise, if the calculated z-score is within the range of the critical values, we fail to reject the null hypothesis.
Once we have the critical value(s), we can compare them with the calculated z-score to make a decision.
By following these steps, the hypothesis test can be accomplished, and a conclusion can be drawn based on the comparison of the test statistic and critical value(s).