Scores on a national exam assume a normal distribution with a population mean of 50 and population standard deviation of 10 points. A sample of 36 exam scores was selected from UCA with a mean score of 58 points. Using a significance level of 0.01, determine whether scores on the exam have actually increased, statistically. What is the null hypothesis?
Ho: µ = sample mean
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In this case, the null hypothesis would be that there is no significant difference in the exam scores before and after, or in other words, the mean score has not increased.
To determine whether scores on the exam have actually increased statistically, we can perform a hypothesis test. Here's how you can do it step by step:
Step 1: State the null hypothesis (H0) and alternative hypothesis (H1).
- Null hypothesis: The mean score on the exam has not increased (μ = 50).
- Alternative hypothesis: The mean score on the exam has increased (μ > 50).
Step 2: Choose the significance level (α). In this case, the significance level is given as 0.01, which means we want to be 99% confident in our decision.
Step 3: Calculate the test statistic. Since we know the population standard deviation (σ) and the sample size (n = 36), we can use the z-test.
- Z = (x̄ - μ) / (σ / √n)
- Z = (58 - 50) / (10 / √36)
- Z = 8 / (10 / 6)
- Z = 4.8
Step 4: Determine the critical value. We need to compare the calculated test statistic (Z) with the critical value from the standard normal distribution table based on the significance level (α = 0.01). Since we have a one-tailed test (μ > 50), we need to find the critical value for the upper tail.
- Critical value = Z(critical)
- Critical value = Z(0.99)
- From the standard normal distribution table, Z(0.99) is approximately 2.33.
Step 5: Make a decision. Compare the calculated test statistic (Z) with the critical value (Z(critical)). If the calculated test statistic is greater than the critical value, we reject the null hypothesis (H0) and conclude that there is statistical evidence to suggest that scores on the exam have actually increased.
- Z > Z(critical)
- 4.8 > 2.33
Since 4.8 is greater than 2.33, we reject the null hypothesis and conclude that scores on the exam have actually increased, statistically.