Question

The average house has 14 paintings on its walls. Is the mean different 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.

15, 13, 14, 15, 13, 11, 13, 12, 15, 14, 11, 13, 12

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

a. For this study, we should use Select an answer b. The null and alternative hypotheses would be: Ho: ? Select an answer H1: ? Select an answer c. The test statistic ? = (please show your answer to 3 decimal places.) d. The p-value = (Please show your answer to 4 decimal places.) e. The p-value is ? a f. Based on this, we should Select an answer the null hypothesis. g. Thus, the final conclusion is that ... The data suggest the population mean is not significantly different from 14 at a = 0.10, so there is sufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is equal to 14. The data suggest that the population mean number of paintings that are in teachers' houses is not significantly different from 14 at a = 0.10, so there is insufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is different from 14. The data suggest the populaton mean is significantly different from 14 at a = 0.10, so there is sufficient evidence to conclude that the population mean number of paintings that are in teachers' houses is different from 14. h. Interpret the p-value in the context of the study. If the population mean number of paintings that are in teachers' houses is 14 and if you survey another 13 teachers then there would be a 5.07% chance that the population mean would either be less than 13 or greater than 15. There is a 5.07% chance of a Type I error. If the population mean number of paintings that are in teachers' houses is 14 and if you survey another 13 teachers, then there would be a 5.07% chance that the sample mean for these 13 teachers would either be less than 13 or greater than 15. There is a 5.07% chance that the population mean number of paintings that are in teachers' houses is not equal to 14.

Answer 1

Solution(a)
In this question we will use t test
Solution(b)
Given in the question
Null hypothesis H0: $\mu$ = 14
Alternative hypothesis Ha: $\mu$ $\neq$ 14
No. of sample = 13
Sample mean = (15+13+14+15+13+11+13+12+15+14+11+13+12)/13 = 171/13 = 13.15
Sample standard deviation can be calculated as
Sample standard deviation = sqrt((Summation(Xi-mean)^2)/(N-1))

X	Xi-mean	(Xi-mean)^2
15	1.85	3.4225
13	-0.15	0.0225
14	0.85	0.7225
15	1.85	3.4225
13	-0.15	0.0225
11	-2.15	4.6225
13	-0.15	0.0225
12	-1.15	1.3225
15	1.85	3.4225
14	0.85	0.7225
11	-2.15	4.6225
13	-0.15	0.0225
12	-1.15	1.3225
	Summation(Xi-mean)^2	23.6925

Standard deviation = sqrt(23.6925/12) = 1.41
Solution(c)
test statistic can be calculated as
Test stat = (Xbar - $\mu$ )/S/sqrt(n) = (13.15-14)/1.41/sqrt(13) = -2.17
Solution(d)
From t table we found p-value at df = 12 and this is two tailed test 0.0507
P-value = 0.0507
Solution(e)
Here alpha =0.10
So P-value is less than alpha vlaue i.e. (0.0507<0.10)
Solution(f)
Based on this we will reject Null hypothesis.
Solution(g)
Its correct answer is C. i.e. the data suggest the population mean is significantly different from 14 at alpha = 0.10, so there is sufficient evidence to conclude that the population mean number of painting that are in teachers houses is different from 14.
Solution(h)
Its answer is D i.e. there is a 5.07% chance that the population mean number of paintings that are in teacher's houses is not equal to 14.