Question

3. A parenting magazine reports that the mean number of phone calls that teenage girls make per
night is at least four (4). For a science fair project, a student sets out to prove the magazine
wrong. The student claims that the average number of phone calls that teenage girls in their
area make is less than four. The student collects information from a simple random sample
of 25 teenage girls from their high school, and calculates a mean of 3.4 calls per night with a
sample standard deviation of 0.9 calls per night. Test the student’s claim at the 0.01 level of
significance.
(a) (3 points) State the null and alternative hypotheses H0 and Ha.
(b) (2 points) Identify which distribution to use for the test statistic. If applicable, calculate
the number df of degrees of freedom; if not applicable, then say so.
(c) (2 points) Compute the value of the test statistic z, t, or χ
2
.
(d) (2 points) Compute the p-value associated to the test statistic.
(e) (1 point) Choose whether or not to reject the null hypothesis H0.

4. (3 points) A new whitening toothpaste advertises that it whitens teeth up to three shades
whiter. The product has been so successful that the company wants to change its slogan to say
“more than three shades whiter”. Before changing the slogan, the company’s executives want
to test this new claim with an hypothesis test. According to the sample that they obtained,
the decision is to reject the null hypothesis. If, in reality, the mean number of shades that the
toothpaste whitens teeth is three shades whiter, was an error made in the hypothesis testing
process? If so, of which type?

Answer 1

3.

Given that,
population mean(u)=4
sample mean, x =3.4
standard deviation, s =0.9
number (n)=25
null, Ho: μ=4
alternate, H1: μ<4
level of significance, α = 0.01
from standard normal table,left tailed t α/2 =2.492
since our test is left-tailed
reject Ho, if to < -2.492
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =3.4-4/(0.9/sqrt(25))
to =-3.3333
| to | =3.3333
critical value
the value of |t α| with n-1 = 24 d.f is 2.492
we got |to| =3.3333 & | t α | =2.492
make decision
hence value of | to | > | t α| and here we reject Ho
p-value :left tail - Ha : ( p < -3.3333 ) = 0.00139
hence value of p0.01 > 0.00139,here we reject Ho
ANSWERS
---------------
a.
null, Ho: μ=4
alternate, H1: μ<4
b.
standard normal distribution
c.
test statistic: -3.3333
critical value: -2.492
e.
decision: reject Ho
d.
p-value: 0.00139
we have enough evidence to support the claim that the average number of phone calls that teenage girls in their
area make is less than four.
4.
the company’s executives want
to test this new claim with an hypothesis test. According to the sample that they obtained,
the decision is to reject the null hypothesis. If, in reality, the mean number of shades that the
toothpaste whitens teeth is three shades whiter,
Type 1 error is possible because it reject the null hypothesis