Question

The average house has 10 paintings on its walls. Is the mean smaller for houses owned by teachers? The data show the results of a survey of 13 teachers who were asked how many paintings they have in their houses. Assume that the distribution of the population is normal.

8, 9, 10, 10, 8, 10, 9, 8, 11, 10, 7, 7, 7

What can be concluded at the α = 0.10 level of significance?

1. For this study, we should use Select an answer z-test for a population proportion or t-test for a population mean
2. The null and alternative hypotheses would be:

H0:? p μ  Select an answer = > ≠ <       

H1:? p μ  Select an answer < > ≠ =    

3. The test statistic? z t  = ( ) (please show your answer to 3 decimal places.)
4. The p-value = ( ) (Please show your answer to 4 decimal places.)
5. The p-value is ? > ≤  α
6. Based on this, we should Select an answer accept, fail to reject, reject the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest that the population mean number of paintings that are in teachers' houses is not significantly less than 10 at αα = 0.10, so there is insufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is less than 10.
   • The data suggest the population mean is not significantly less than 10 at α = 0.10, so there is sufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is equal to 10.
   • The data suggest the population mean is significantly less than 10 at α = 0.10, so there is sufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is less than 10.
8. Interpret the p-value in the context of the study.
   • If the population mean number of paintings that are in teachers' houses is 10 and if you survey another 13 teachers, then there would be a 0.34% chance that the sample mean for these 13 teachers would be less than 8.77.
   • If the population mean number of paintings that are in teachers' houses is 10 and if you survey another 13 teachers, then there would be a 0.34% chance that the population mean number of paintings that are in teachers' houses would be less than 10.
   • There is a 0.34% chance of a Type I error.
   • There is a 0.34% chance that the population mean number of paintings that are in teachers' houses is less than 10.
9. Interpret the level of significance in the context of the study.
   • If the population mean number of paintings that are in teachers' houses is 10 and if you survey another 13 teachers, then there would be a 10% chance that we would end up falsely concluding that the population mean number of paintings that are in teachers' houses is less than 10.
   • There is a 10% chance that teachers are so poor that they are all homeless.
   • There is a 10% chance that the population mean number of paintings that are in teachers' houses is less than 10.
   • If the population mean number of paintings that are in teachers' houses is less than 10 and if you survey another 13 teachers, then there would be a 10% chance that we would end up falsely concluding that the population mean number of paintings that are in teachers' houses is equal to 10.

Answer 1

Given that,
population mean(u)=10
sample mean, x =8.769
standard deviation, s =1.309
number (n)=13
null, Ho: μ=10
alternate, H1: μ<10
level of significance, α = 0.1
from standard normal table,left tailed t α/2 =1.356
since our test is left-tailed
reject Ho, if to < -1.356
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =8.769-10/(1.309/sqrt(13))
to =-3.391
| to | =3.391
critical value
the value of |t α| with n-1 = 12 d.f is 1.356
we got |to| =3.391 & | t α | =1.356
make decision
hence value of | to | > | t α| and here we reject Ho
p-value :left tail - Ha : ( p < -3.3907 ) = 0.00268
hence value of p0.1 > 0.00268,here we reject Ho
ANSWERS
---------------
a.
T test for single mean with unknown population standard deviation
b.
null, Ho: μ=10
alternate, H1: μ<10
c.
test statistic: -3.391
critical value: -1.356
p-value: 0.00268
p value is less than alpha value
d.
decision: reject Ho
e.
we have enough evidence to support the claim that the mean smaller for houses owned by teachers
f.
If the population mean number of paintings that are in teachers' houses is 10 and if you survey another 13 teachers, then there would be a 0.34% chance that
the population mean number of paintings that are in teachers' houses would be less than 10.
g.
There is a 10% chance that the population mean number of paintings that are in teachers' houses is less than 10.