the average math sat score is 500 with a standard deviation of 100. a particular high school claims that its students have usually high math SAT scores a random sampel of 50 students from this school was selected, and the mean math sat score was 530 . Is the high school justified in its claim?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Is it smaller than your alpha error?
To determine whether the high school's claim is justified or not, we can use statistical hypothesis testing. Specifically, we can perform a one-sample t-test.
Here's how to do it step by step:
Step 1: Define the hypotheses.
- Null Hypothesis (H0): The mean math SAT score of the high school's students is equal to the average math SAT score of 500.
- Alternative Hypothesis (H1): The mean math SAT score of the high school's students is greater than the average math SAT score of 500.
Step 2: Set the significance level (alpha).
- The significance level determines the probability of rejecting the null hypothesis when it is actually true. Let's assume a significance level of 0.05, which is a commonly used value.
Step 3: Collect the necessary information.
- Population mean (𝜇): 500 (provided in the question)
- Population standard deviation (𝜎): 100 (provided in the question)
- Sample mean (x̄): 530 (provided in the question)
- Sample size (n): 50 (provided in the question)
Step 4: Calculate the test statistic.
- The test statistic can be calculated using the formula: t = (x̄ - 𝜇) / (𝜎 / √n), where t represents the test statistic.
- Plugging in the values: t = (530 - 500) / (100 / √50)
Step 5: Determine the critical value or p-value.
- For a one-tailed test (since the claim is about being "usually high"), we need to find the critical value at a significance level of 0.05. This can be done using a t-distribution table or a statistical software.
- Alternatively, you can find the p-value associated with the test statistic using the t-distribution. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.
Step 6: Make the decision.
- If the test statistic is greater than the critical value (or if the p-value is less than the significance level), we reject the null hypothesis.
- If the test statistic is less than the critical value (or if the p-value is greater than the significance level), we fail to reject the null hypothesis.
Based on this information, perform the calculations and make your decision accordingly.