The SAT scores are normally distributed. A simple random sample of 100 SAT score

Question

The SAT scores are normally distributed. A simple random sample of 100 SAT scores has a sample mean of 1500 and a sample standard deviation of 300.

(a) What distribution will you use to determine the critical value for a confidence interval estimate of the mean SAT score? Why?

(b) Construct a 90% confidence interval estimate of the mean SAT score. (Show work and round the answer to two decimal places)

(c) Is a 95% confidence interval estimate of the mean SAT score wider than the 90% confidence interval estimate you got from part (b)? Why? [You don’t have to construct the 95% confidence interval]

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Explanation / Answer

Here we have given that the SAT scores are normally distributed. A simple random sample of 100 SAT scores has a sample mean of 1500 and a sample standard deviation of 300.

(a) What distribution will you use to determine the critical value for a confidence interval estimate of the mean SAT score? Why?

Here we use t-distribution to find the critical value for a confidence interval estimate of the mean SAT score.

Because population standard deviation is unknown.

(b) Construct a 90% confidence interval estimate of the mean SAT score.

Confidence interval we can find by using TI-83 calculator.

steps :

STAT --> TESTS --> 8:TInterval --> ENTER --> Highlight on Stats --> ENTER --> Input all the values -->C-level : 0.90 --> Calculate --> ENTER

90% confidence  interval estimate of the mean SAT score is (1450.2, 1549.8).

(c) Is a 95% confidence interval estimate of the mean SAT score wider than the 90% confidence interval estimate you got from part (b)? Why?

Higher confidence (95%) has wider confidence interval whereas lower confidence (90%) has narrower  confidence interval.

But your probability of encountering an observation with that exact same value would be far lower than finding one within any particular sample based CI.

95% confidence interval is wider than a 90%.

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