A study considers if the mean score on a college entrance exam for students in 2005 is any different from the mean score of 501 for students who took the same exam in 1975. Let μ represent the mean score for all students who took the exam in 2005. For a random sample of 40,000 students who took the exam in 2005, x = 499 and s = 100.(a) Find the test statistic.(b) Find the P-value for testing H0: μ = 501 against Ha: μ ≠ 501.a. 0.00006b. 0.03456c. 0.06845d. 0.14282(c) Explain why the test result is statistically significant but not practically significant.a. This result is statistically significant because the P-value is very small, but it is not practically significant because the null hypothesis mean of 501 is not equal to the sample mean of 499.b.This result is statistically significant because the P-value is not very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.c. This result is statistically significant because the P-value is very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.

This result is statistically significant because the p-value is very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.

Step-by-step explanation:

a)

The test statistic z is given by

[tex]\bf z=\frac{\bar x-\mu}{s/\sqrt{n}}[/tex]

where

[tex]\bf \bar x[/tex]= 499 the mean for the random sample

[tex]\bf \mu[/tex]= 501 the mean in 1975

s = 100 the standard deviation of the sample

n = 40,000 the sample size

So,

[tex]\bf z=\frac{499-501}{100/\sqrt{40,000}}=-4[/tex]

b)

The p-value would be the area under the Normal N(0,1) outside the interval [-4, 4]. This area equals 0.00006

This result is statistically significant because the p-value is very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.

joaobezerra joaobezerra · Answer 2 · 2021-10-28T15:03:19+02:00

Testing the hypothesis, using the t-distribution, we have that:

a) The test statistic is t = 4.

b)

a. 0.00006

c)

c. This result is statistically significant because the P-value is very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.

The hypothesis that are tested are:

[tex]H_0: \mu = 501[/tex]

[tex]H_1: \mu \neq 501[/tex]

We have the standard deviation for the sample, thus, the t-distribution is used to solve this question. The test statistic is:

[tex]t = \frac{x - \mu}{\frac{s}{\sqrt{n}}}[/tex]

Item a:

In this problem, we have that: [tex]x = 499, \mu = 501, s = 100, n = 40000[/tex].

The value of the test statistic is:

[tex]t = \frac{x - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{499 - 501}{\frac{100}{\sqrt{40000}}}[/tex]

[tex]t = 4[/tex]

The test statistic is t = 4.

Item b:

The p-value is found using a two-tailed test(as we are testing if the mean is different of a value), with t = 4 and 40000 - 1 = 39999 df.

Using a calculator, this p-value is 0.00006, option a.

Item c:

The smaller the p-value, the more statistically significant it is, thus, the correct option is:

c. This result is statistically significant because the P-value is very small, but it is not practically significant because the sample mean of 499 is very close to the null hypothesis mean of 501.

A similar problem is given at https://brainly.com/question/24146681