The average math SAT score is 500 with a standard deviation of 100. A particular high school claims that its students have unusually high math SAT scores. A random sample of 50 students from this school was selected, and the mean math SAT score was 530. Is the high school justified in its claim? Explain.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
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 related to your Z score.
To determine whether the high school is justified in its claim of unusually high math SAT scores, we can use statistical analysis.
First, let's calculate the standard error of the mean using the formula:
Standard Error of the Mean = Standard Deviation / √(Sample Size)
Standard Error of the Mean = 100 / √(50)
Standard Error of the Mean = 14.14
Next, let's calculate the z-score, which measures how many standard errors the sample mean is away from the population mean.
z-score = (Sample Mean - Population Mean) / Standard Error of the Mean
z-score = (530 - 500) / 14.14
z-score ≈ 2.12
Using a z-table or calculator, we can find that the proportion of the population falling to the right of a z-score of 2.12 is approximately 0.0174.
This means that only 1.74% of the population would have a mean math SAT score higher than the sample mean of 530 if the high school's claim were not true.
Since this is a very small proportion, we can conclude that the high school's claim of unusually high math SAT scores is justified based on this sample.
To determine if the high school is justified in its claim, we need to perform a hypothesis test.
Step 1: State the hypotheses.
The null hypothesis (H₀) is that the high school's students have an average math SAT score equal to the average math SAT score of 500. The alternative hypothesis (Hₐ) is that the high school's students have an average math SAT score greater than 500.
Step 2: Set the significance level.
This step involves setting the level of significance, denoted as α. The significance level determines how rare the sample results need to be in order to reject the null hypothesis. Let's assume a significance level of 0.05 (typical choice).
Step 3: Calculate the test statistic.
In this case, we will use a one-sample t-test since the population standard deviation is unknown. The formula for the test statistic is:
t = (x̄ - μ₀) / (s / √n)
Where:
x̄ is the sample mean (530),
μ₀ is the hypothesized population mean (500),
s is the sample standard deviation (100),
n is the sample size (50).
Plugging in the values, we get:
t = (530 - 500) / (100 / √50) = 3.54 (rounded to two decimal places).
Step 4: Determine the critical value or p-value.
Since the alternative hypothesis is that the average math SAT score is greater than 500, we need to find the critical t-value in the upper tail of the t-distribution with 49 degrees of freedom for a one-sided test at the 0.05 significance level. Using a t-table or statistical software, we find that the critical value is approximately 1.67.
Step 5: Make a decision.
If the test statistic t is greater than the critical value (3.54 > 1.67), we reject the null hypothesis. Alternatively, if the p-value is less than the significance level (0.05), we also reject the null hypothesis.
Step 6: State the conclusion.
Since the test statistic (3.54) is greater than the critical value (1.67) and the p-value is much smaller than 0.05, we reject the null hypothesis. This means that the high school's claim of unusually high math SAT scores is justified.
In conclusion, based on the sample data and hypothesis test, we have sufficient evidence to support the high school's claim that its students have unusually high math SAT scores.