Question

1. The College Board wants to calibrate the difficulty of the math SAT test. Difficulty is interpreted as the population mean score. The College Board’s statisticians administer the test to 41 randomly-selected students. The mean score in the sample is 534. The statisticians are willing to assume that scores on the math SAT are approximately normally distributed, and that the population standard deviation is 100 points.

The SAT authors are aiming for a population mean score of 525. They use their sample to test the null hypothesis that the population mean is 525, against the alternative hypothesis that it is not 525, using a 1% level of significance.

Which sampling distribution is the basis for this hypothesis test?  Letter (see multiple choices in the instructions)

2. Begin by using the rejection region approach. What is the form of the rejection region?  Choose the letter that represents your choice.

3. Give the value of the critical value (cv) if the rejection region has one critical value or the value of the smaller critical value (cv₁) if the rejection region has two critical values.  One decimal

4. Give the value of the larger critical value (cv₂) if the rejection region has two critical values. If the rejection region has only one critical value, enter NA.  One decimal

5. What is the conclusion of the test?

Fail to reject H0 (accept H0)

Reject H0 in favor of HA

6. What is the probability of Type II error if the population mean is 510?  Four decimals

7. Continued. What is the power of the test if the population mean is 510?  Four decimals

8. What is the probability of Type II error if the population mean is 540?  Four decimals

9. Continued. What is the power of the test if the population mean is 540?  Four decimals

Answer 1

1.

Sampling distribution: Normal distribution(z test)

2.

Rejection region:

Reject Ho if z < -z_crit or if z > z_crit

3.

Smaller Critical value, z_crit = ABS(NORM.S.INV(0.01/2)) = -2.6

4.

Larger Critical value, z_crit = ABS(NORM.S.INV(0.01/2)) = 2.6

5.

Test statistic:

z = (x̅- µ)/(σ/√n) = (534 - 525)/(100/√41) = 0.5763

Decision:

Fail to reject H0 (accept H0)

6.

µ = 525, σ = 100, n = 41
Critical value, z_c = NORM.S.INV(0.01/2) = 2.576
Z = (X̅ - µ) / (σ/√n)
Reject if Z ≤ -2.576 or if Z ≥ 2.576
Reject if Z ≤ -2.576
X̅1 ≤ 525 + (-2.576) * 100/√41
X̅1 ≤ 484.7723
or
Reject if Z ≥ 2.576
X̅2 ≥ 525 + (2.576) * 100/√41
X̅2 ≥ 565.2277
The researchers will reject H₀ if the sample mean is less than 484.7723 or greater than 565.2277

If true µ₁ = 510, then type II error, β =

β = P(484.7723 < X̅ < 565.2277 | µ₁ = 510)

= P( (484.7723-510)/(100/√41) ) < Z < (565.2277-510)/(100/√41) )

= P(-1.6154 < Z < 3.5363)

=P(Z < 3.5363) - P(Z < -1.6154)

Using excel function:

= NORM.S.DIST(3.5363, 1) - NORM.S.DIST(-1.6154, 1)

= 0.9467

7.

Power = 1- Type II error

= 1-0.9467

= 0.0533

8.

If true µ₁ = 540, then type II error, β =

β = P(484.7723 < X̅ < 565.2277 | µ₁ = 540)

= P( (484.7723-540)/(100/√41) ) < Z < (565.2277-540)/(100/√41) )

= P(-3.5363 < Z < 1.6154)

=P(Z < 1.6154) - P(Z < -3.5363)

Using excel function:

= NORM.S.DIST(1.6154, 1) - NORM.S.DIST(-3.5363, 1)

= 0.9467

9.

Power = 1- Type II error

= 1-0.9467

= 0.0533