A recent national survey reported that the general population gave the president an average rating of m= 60 on a scale of 1 to 100 with 1 being most unfavorable, and 100 most favorable. A reseracher at IC hypothesizes that IC students are more critcal of the president than the general population. She collects data on 10 IC students (below). at significance level of alpha=.05 test the one-tailed hypothesis that IC students are more critical than the population.

sample ratings of president
44 50 24 45 39 57 25 90 78 54

With such a small sample, you need to use a t-test with 9 df., t<2.262 to be significant one-tailed at P = .05.

The process is similar to the one given in your later post.

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

To test the hypothesis that IC students are more critical of the president than the general population, we can conduct a one-tailed hypothesis test. Here's how you can perform the test:

Step 1: Define the null and alternative hypotheses:
- Null hypothesis (H0): The average rating of IC students is not significantly different from the general population (µ = 60).
- Alternative hypothesis (Ha): The average rating of IC students is significantly lower than the general population (µ < 60).

Step 2: Calculate the sample mean and standard deviation:
From the given data, we find the sample mean (x̄) by summing up the ratings and dividing by the sample size:
x̄ = (44 + 50 + 24 + 45 + 39 + 57 + 25 + 90 + 78 + 54) / 10 = 506 / 10 = 50.6

Next, calculate the sample standard deviation (s) using the formula:
s = √[(Σ(xi - x̄)²) / (n - 1)]

In this case:
s = √[( (44 - 50.6)² + (50 - 50.6)² + ... + (54 - 50.6)²) / 9] = √[1435.6 / 9] ≈ √159.51 ≈ 12.63

Step 3: Calculate the test statistic:
The test statistic for this case is the t-value, which is calculated as:
t = (x̄ - µ) / (s / √n)

In this case:
t = (50.6 - 60) / (12.63 / √10) ≈ -9.4 / 3.99 ≈ -2.35

Step 4: Determine the critical value:
With a significance level (α) of 0.05 and a one-tailed test, we need to find the critical value from the t-distribution tables. Since degrees of freedom (df) = n - 1 = 9, the critical value for α = 0.05 is approximately -1.83 (from the table or using statistical software).

Step 5: Make a decision:
Compare the calculated t-value (-2.35) with the critical value (-1.83). If the calculated t-value is smaller than the critical value, we reject the null hypothesis in favor of the alternative hypothesis.

Since -2.35 < -1.83, we can reject the null hypothesis.

Step 6: Draw a conclusion:
Based on the sample data, there is sufficient evidence to conclude that IC students are significantly more critical of the president than the general population.

Remember, hypothesis testing is a statistical method that allows us to make conclusions about a population based on sample data. It helps to provide evidence for or against a proposed hypothesis.