Question

A math teacher claims that she has developed a review course that increases the scores of students on the math portion of a college entrance exam. Based on data from the administrator of the​ exam, scores are normally distributed with mu equals510. The teacher obtains a random sample of 1800 ​students, puts them through the review​ class, and finds that the mean math score of the 1800 students is 515 with a standard deviation of 111. Complete parts​ (a) through​ (d) below.

​(a) State the null and alternative hypotheses. Let mu be the mean score. Choose the correct answer below.

A.

Upper H 0 : mu equals 510​, Upper H 1 : mu not equals 510

B.

Upper H 0 : mu greater than 510​, Upper H 1 : mu not equals 510

C.

Upper H 0 : mu less than 510​, Upper H 1 : mu greater than 510

D.

Upper H 0 : mu equals 510​, Upper H 1 : mu greater than 510

​(b) Test the hypothesis at the alpha equals0.10 level of significance. Is a mean math score of 515 statistically significantly higher than 510​? Conduct a hypothesis test using the​ P-value approach.

Find the test statistic.

t 0equals

nothing

​(Round to two decimal places as​ needed.)

Find the​ P-value.

The​ P-value is

nothing.

​(Round to three decimal places as​ needed.)

Is the sample mean statistically significantly​ higher?

Yes

No

​(c) Do you think that a mean math score of 515 versus 510 will affect the decision of a school admissions​ administrator? In other​ words, does the increase in the score have any practical​ significance?

​No, because the score became only 0.98​% greater.

​Yes, because every increase in score is practically significant.

​(d) Test the hypothesis at the alphaequals0.10 level of significance with nequals375 students. Assume that the sample mean is still 515 and the sample standard deviation is still 111. Is a sample mean of 515 significantly more than 510​? Conduct a hypothesis test using the​ P-value approach.

Find the test statistic.

t 0equals

nothing

​(Round to two decimal places as​ needed.)

Find the​ P-value.

The​ P-value is

nothing.

​(Round to three decimal places as​ needed.)

Is the sample mean statistically significantly​ higher?

Yes

No

What do you conclude about the impact of large samples on the​ P-value?

A.

As n​ increases, the likelihood of rejecting the null hypothesis increases.​ However, large samples tend to overemphasize practically significant differences.

B.

As n​ increases, the likelihood of not rejecting the null hypothesis increases.​ However, large samples tend to overemphasize practically insignificant differences.

C.

As n​ increases, the likelihood of rejecting the null hypothesis increases.​ However, large samples tend to overemphasize practically insignificant differences.

D.

As n​ increases, the likelihood of not rejecting the null hypothesis increases.​ However, large samples tend to overemphasize practically significant differences.

Click to select your answer(s).

Answer 1

Given that,
population mean(u)=510
sample mean, x =515
standard deviation, s =111
number (n)=1800
null, Ho: μ=510
alternate, H1: μ>510
level of significance, alpha = 0.1
from standard normal table,right tailed t alpha/2 =1.28
since our test is right-tailed
reject Ho, if to > 1.28
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =515-510/(111/sqrt(1800))
to =1.911
| to | =1.911
critical value
the value of |t alpha| with n-1 = 1799 d.f is 1.28
we got |to| =1.911 & | t alpha | =1.28
make decision
hence value of | to | > | t alpha| and here we reject Ho
p-value :right tail - Ha : ( p > 1.9111 ) = 0.02808
hence value of p0.1 > 0.02808,here we reject Ho
ANSWERS
---------------
a.
null, Ho: μ=510
alternate, H1: μ>510
test statistic: 1.911
critical value: 1.28
decision: reject Ho
p-value: 0.02808
b.
yes,
we have enough evidence to support the claim that mean math score of 515 statistically significantly higher than 510​
c.
Given that,
population mean(u)=510
sample mean, x =515
standard deviation, s =111
number (n)=1800
null, Ho: μ=510
alternate, H1: μ!=510
level of significance, alpha = 0.1
from standard normal table, two tailed t alpha/2 =1.65
since our test is two-tailed
reject Ho, if to < -1.65 OR if to > 1.65
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =515-510/(111/sqrt(1800))
to =1.911
| to | =1.911
critical value
the value of |t alpha| with n-1 = 1799 d.f is 1.65
we got |to| =1.911 & | t alpha | =1.65
make decision
hence value of | to | > | t alpha| and here we reject Ho
p-value :two tailed ( double the one tail ) - Ha : ( p != 1.9111 ) = 0.0562
hence value of p0.1 > 0.0562,here we reject Ho
ANSWERS
---------------
null, Ho: μ=510
alternate, H1: μ!=510
test statistic: 1.911
critical value: -1.65 , 1.65
decision: reject Ho
p-value: 0.0562
​Yes, because every increase in score is practically significant.
d.
Given that,
population mean(u)=510
sample mean, x =515
standard deviation, s =111
number (n)=375
null, Ho: μ=510
alternate, H1: μ>510
level of significance, alpha = 0.1
from standard normal table,right tailed t alpha/2 =1.28
since our test is right-tailed
reject Ho, if to > 1.28
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =515-510/(111/sqrt(375))
to =0.872
| to | =0.872
critical value
the value of |t alpha| with n-1 = 374 d.f is 1.28
we got |to| =0.872 & | t alpha | =1.28
make decision
hence value of |to | < | t alpha | and here we do not reject Ho
p-value :right tail - Ha : ( p > 0.8723 ) = 0.1918
hence value of p0.1 < 0.1918,here we do not reject Ho
ANSWERS
---------------
null, Ho: μ=510
alternate, H1: μ>510
test statistic: 0.872
critical value: 1.28
decision: do not reject Ho
p-value: 0.1918
No,
the sample mean statistically significantly​ higher
option:
D.
As n​ increases, the likelihood of not rejecting the null hypothesis increases.
​ However, large samples tend to overemphasize practically significant differences