Question

Ch.15-18 Due: Nov. 7 Please answer the following questions below. You must include your work or no credit will be given. Keep at least 4 decimal places. The National Collegiate Athletic Association (NCAA) requires Division II athletes to score at least 820 on the combined mathematics and reading parts of the SAT in order to compete in their first college year. The scores of the 1.5 million high school seniors taking the SAT last year are approximately Normal with mean 1026 and standard deviation 209. For an SRS size 200 :

(1) Find the mean and standard deviation of Mean_______ Standard deviation_____

(2) What is the distribution of "x bar?

(3) What is the probability that a sample mean value exceeds 1028? _______

(4) The highest 2.5% of sample mean value are higher than ______

(5) Has the average score increased since last year? To answer this, do the followings: I. A SRS size 200 gives a sample mean of 1028. State hypotheses, find the test statistic, pvalue, and express your conclusion using a significance level of α=2.5%.

a. Hypotheses (Use notations)

b. test statistic

c. pvalue

d. Conclusion? Include practical terms.

II. Find a 95% confidence interval for the mean SAT score.

(6) What sample size would you need to ensure a margin of error is 10 for a 95% confidence

Answer 1

1. As sample size is sufficient large, so as per central limit theorem, distribution of sample mean is also normal as per central limit theorem

So sample mean is $\overline{x}=\mu=1026$ and standard deviation is $\sigma_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{209}{\sqrt{200}}=14.7785$

2, Distribution of sample mean as per 1, is normal

3. Now we need to find $P(\overline{x}>1028)$

As sample mean is normal, we can convert sample mean to z

$P(\overline{x}>1028)=P(z>\frac{1028-1026}{14.7785})=P(z>0.14)=0.4443$

4.Here we need to find $P(\overline{X}>\overline{x})=0.025$

Using z table we get $P(z>1.96)=0.025$

So $z=1.96=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$

So $\overline{x}=1.96*\frac{s}{\sqrt{n}}+\mu=1.96*\frac{209}{\sqrt{200}}+1026=1054.9659$