Final answer:
The math SAT scores are initially distributed normally with a mean (μ) of 500 and a variance (σ2) of 10,000. The evening school claims to improve scores by 30 points, resulting in a new mean (μ′) of 530. However, the variance (σ2) remains unchanged. The new distribution for the improved scores is N(530,100²), indicating a normal distribution with a mean of 530 and a standard deviation of 100.
Step-by-step explanation:
Certainly! Let's break down the problem:
Given:
Mean (μ): 500
Variance (σ2): 10,000
The standard deviation (σ) is the square root of the variance, so in this case, σ= √10,000 =100.
Now, the evening school claims to improve students' scores by 30 points. If X is a random variable representing the SAT scores, the new mean (μ′ ) would be the old mean plus the improvement:
μ′ =μ+improvement=500+30=530.
The variance remains the same because we're assuming the improvement doesn't affect the variability of scores.
The new standard deviation (σ′) is still 100.
So, the new distribution for the improved scores is N(530,100²).
Now, if you have specific questions or if there's a particular aspect you'd like to explore (such as probability calculations or further analysis), feel free to specify!
Your correct question is: Problem 1. Suppose that math SAT scores are distributed normally with mean 500 and variance 10,000 . An evening school advertises that it can improve students' scores by 30 points (roughly a third of a standard deviation) if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.) The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: Let M denote the math SAT score of a student after attending the class and suppose M is normally distributed with (unknown) mean μM and (known) variance 10,000. Then the testing problem is H0:
μM =500 vs. H1 μM>500. (a) Sketch the two distributions (of M ) under the null hypothesis and when μ M=530. (b) The consumer protection agency wants to evaluate this claim by sending 50 students (assumed to constitute a random sample from the population) to attend classes. One of the students becomes sick during the course and drops out. What is the distribution of the average score of the remaining 49 students under the null, and when μ M=530 ? (c) Assume that after graduating from the course, the 49 participants take the SAT test and score an average of 520 . Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis? (d) What would be the critical value under the null hypothesis if the size of your test were 5% ? (e) Given this critical value, what is the power of the test when μM=530 ? What options does the statistician have for increasing the power in this situation?