Question

A) The following observations are obtained when a random sample is drawn from a normally distributed population: [7.472, 37.89, 21.32, 25.99, 15.8]

Use this information to test the null hypothesis H0:σ2=60 against the alternative hypothesis HA:σ2>60.

a) What is the value of the test statistic χ2? Round your response to at least 3 decimal places.

b) The P-value falls within which one of the following ranges:

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.025 < P-value < 0.05
D		0.01 < p-value < 0.025
E		P-value < 0.01

c) What conclusion can be made, at the 10% level of significance?

A		There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is greater than 60.
B		There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 60.

B)  The following observations are obtained when a random sample is drawn from a normally distributed population: [9.939, 36.83, 22.03, 29.47, 19.94]

Use this information to test the null hypothesis H0:σ2=120 against the alternative hypothesis HA:σ2<120.

a) What is the value of the test statistic χ2? Round your response to at least 3 decimal places.

b) The P-value falls within which one of the following ranges:

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.025 < P-value < 0.05
D		0.01 < P-value < 0.025
E		P-value < 0.01

c) What conclusion can be made, at the 10% level of significance?

A		There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is less than 120.
B		There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 120.

C) The amount a person is willing to pay for a 45 inch LED TV used to be about $500. A manufacturer wonders if this value is lower now. The manufacturer takes a random sample of 30 people and finds they are willing to pay an average of $469 with a standard deviation of $61. At the .01 significance level, conduct a full and appropriate hypothesis test for the manufacturer.

a) What are the appropriate null and alternative hypotheses?

A		H0:μ=500H1:μ<500
B		H0:x¯=500H1:x¯>500
C		H0:x¯=469H1:x¯>469
D		H0:μ=469H1:μ<469
E		H0:x¯=469H1:x¯<469
F		H0:μ=469H1:μ>469
G		H0:x¯=500H1:x¯<500
H		H0:μ=500H1:μ>500

b) Identify the values given in the problem:

1) =469

2) =500

3) = .01

4) =61

5) = 30

c) Calculate the value of the test statistic.  Round your response to at least 2 decimal places.

d) What is the corresponding P-value for the test statistic? Round your response to at least 4 decimal places.

e) Make a decision:

Since α (<, >, =) P, we (accept, reject) the null hypothesis (H0, H1)

f) Help write a summary of the results of this hypothesis test:

( There is not,There is, We do not whether there is) enough evidence in this sample to conclude the average price (women, this city's population, people, Austin's Population, men) are willing to pay for a 45 inch HD LED TV is (different from, less than, greater than) $500 at the α= (.05, .01, .1) significance level because P= ? .

Answer 1

A) from the data we found sample variance

S²=129.5 , sigma² = 60

H0:- sigma²= 60 vs Ha :- sigma²> 60

Chi square statistics = (n-1)*S² / sigma²

Chi stat = ( 4 * 129.5 ) / 60 = 8.633

P.value = 0.0709 < 0.10

0.05 < P value < 0.10

There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is less than 120.

B) from the data set we found sample variNce

S²=103 , sigma ² = 120 , n=5

Ho :- sigma ² = 120

Ha :- sigma ² < 120

Chi square stat = (4*103)/120 = 3.433

P.value = 0.4881 > 0.10

There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 120.

C)

A] H0: mu=500 .vs H1: mu <500

1)xbar =469

2)mu =500

3) alpha= .01

4) s =61

5) n= 30

Test statistics

T = ( x bar - mu ) / s ÷ sqrt ( n ) = -2.78

P value = p ( t alpha < -2.784 )

P = 0.0047 < 0.01 = alpha

We reject Ho

e) Make a decision:

Since α (>) P, we (reject) the null hypothesis (H0)

f) Help write a summary of the results of this hypothesis test:

( There is)enough evidence in this sample to conclude the average price (people) are willing to pay for a 45 inch HD LED TV is ( less than) $500 at the α= ( .01)significance level because P=0.0047