Question

SAT scores of students at an Ivy League college are distributed with a standard deviation of 250 points. Two statistics students, Raina and Luke, want to estimate the average SAT score of students at this college as part of a class project. They want their margin of error to be no more than 25 points.

(a) Raina wants to use a 90% condence interval. How large a sample should she collect?
Raina should sample at least  people.

(b) Luke wants to use a 99% condence interval. Without calculating the actual sample size, determine whether his sample should be larger or smaller than Raina's, and explain your reasoning.

smaller since Luke has a higher level of confidence in his results than Raina

smaller because higher degrees of confidence require smaller margins of error

larger higher degrees of confidence require larger margins of error

(c) Calculate the minimum required sample size for Luke.
Luke should sample at least  people.

Answer 1

Concepts and reason

The sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample.

The sample size calculation includes margin of error, critical value and standard deviation

Fundamentals

The formula for the sample size is as follows:

Here,

n is the sample size.

${t_{\alpha /2}}$ is the critical value for $\alpha$ level of significance

$\alpha$ is the level of significance.

E is the margin of error.

(a)

The standard deviation of the SAT scores is

The given margin of error value is

Given level of significance is

Compute the sample size required for this study.

From the normal area table values, at 0.10 level of significance the value of is 1.645.

Substitute the values,

(b)

From the known properties, the sample size is directly proportion to the confidence level. If the confidence level of significance increases from 90% to 99% the sample size is also increases.

(c)

The standard deviation of the SAT scores is

The given margin of error value is

Given level of significance is

Compute the sample size required for this study.

From the normal area table values, at 0.10 level of significance the value of is 2.58.

Substitute the values,

Ans: Part a

Raina should sample at least 271 people.

Part b

If the confidence interval increases, the sample size also increases.

Part c

Raina should sample at least 666 people.