Question

Scores on the SAT mathematics section have a normal distribution with mean 4-500 and standard deviation o=100. a. What proportion of students score above a 550 on the SAT mathematics section? Round your answer to 4 decimal places. b. Suppose that you choose a simple random sample of 16 students who took the SAT mathematics section and find the sample mean x of their scores. Which of the following best describes what you would expect? The sample mean will be exactly 500. The sample mean should be close to 500, but not necessarily equal to 500. c. Now suppose that instead of taking just one simple random sample of 16 students who took the SAT mathematics section you take lots of simple random samples of 16 students who took the SAT mathematics section. Which of the following best describes what you would expect? The sample means of the different simple random samples will vary, but they will vary by less than the scores of individual students do. The sample means of the different simple random samples will not vary. The sample means of the different simple random samples will vary by the same amount as the scores of individual students do. The sample means of the different simple random samples will vary, but they will vary by more than the scores of individual students do. Different simple random samples produce different sample means; these different values form the sampling distribution of the sample means. This distribution has the same mean as the population that we're sampling, in this case 500. The standard deviation of a sampling distribution is called the standard error. d. What is the standard error of the sampling distribution of sample mean for samples of size 16 of scores on the SAT mathematics section? SE = When the population distribution is normal, so is the sampling distribution. A Z-score for a sample mean is calculated in the same way as a usual z-score, except with the standard error (SE) replacing the standard deviation: e. What Z-score corresponds to a sample mean of x = 550 ? z= 1. What is the probability that a simple random sample of 16 students taking the SAT mathematics section will have an average score above 550? Round your answer to 4 decimal places.

Answer 1

Solution-A:

X follows normal distribution with mean=500 and standard deviation=100

P(X>550)

Rcode:

library(tigerstats)
pnormGC(bound=550,region="above",mean=500,sd=100,graph=TRUE)

Normal Curve, mean = 500, SD Shaded Area 0.3085 100 500 X density 0.000 0.003 TT

ANSWER:

0.3085

Solutionb:

the sample mean will be close to population mean of 500,but not necessarily equal to 500

MARK OPTION B

Solution-c:

MARK OPTION A

Solution-d:

SE=Sigma/sqrt(sample size)

=100/sqrt(16)

=100/4

=25

SE=25

Solution-e:

z=score-mean/SE

=550-500/25

=50/25

=2

Z=2

Solutionf:

P(Xbar>550)

P(Z>2)

=1-P(Z<2)

=1-0.9772

=0.0228

ANSWER:

0.0228