students who learned English as well as another language simultaneously score worse on the SAT critical Reading exam than the general population of test takers? The mean score among all test takers on SAT Critical Reading exam is 501 A random sample of 100 test takers who learned English as well as another language simultaneously had a mean sat critical reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on SAT Critical Reading Exam
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To determine whether the results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading Exam, we will need to perform a hypothesis test.
Step 1: State the hypotheses:
- Null hypothesis (H0): Students who learn English as well as another language simultaneously have the same mean SAT Critical Reading score as the general population of test takers (µ = 501).
- Alternative hypothesis (Ha): Students who learn English as well as another language simultaneously score lower on the SAT Critical Reading Exam compared to the general population of test takers (µ < 501).
Step 2: Determine the significance level:
We need to choose a significance level, denoted as α, to determine the level of evidence required to reject the null hypothesis. Let's assume a significance level of 0.05 (5%).
Step 3: Collect sample data and calculate the test statistic:
Given the sample data, n = 100, sample mean (x̄) = 485, sample standard deviation (s) = 116.
To calculate the test statistic for a one-sample t-test, we use the formula:
t = (x̄ - µ) / (s / √n)
Substituting the values into the formula, we get:
t = (485 - 501) / (116 / √100) = -16 / (116 / 10) = -1.379
Step 4: Determine the critical value:
Since we are performing a one-sample t-test and our alternative hypothesis is one-tailed (students score lower), we need to find the critical t-value that corresponds to our chosen significance level (α = 0.05) and degrees of freedom (df = n - 1 = 99).
Using a t-table or calculator, the critical t-value for α = 0.05 and df = 99 is approximately -1.660.
Step 5: Compare the test statistic with the critical value:
If the calculated test statistic falls within the critical region (i.e., less than the critical value), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, t = -1.379, which is greater than -1.660. Therefore, we fail to reject the null hypothesis.
Step 6: Interpret the results:
Based on the sample data, there is not enough evidence to suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading Exam compared to the general population of test takers.
It's important to note that the sample mean (485) is lower than the population mean (501), but this difference may be due to random sampling variability rather than a systematic difference between the two groups.