Question

A new kind of typhoid shot is being developed by a medical research team. The old typhoid shot was known to protect the population for a mean time of 36 months, with a standard deviation of 3 months. To test the time variability of the new shot, a random sample of 25 people were given the new shot. Regular blood tests showed that the sample standard deviation of protection times was 1.5 months. Using a 0.05 level of significance, test the claim that the new typhoid shot has a smaller variance of protection times.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 9; H₁: σ² > 9

H_o: σ² < 9; H₁: σ² = 9    

H_o: σ² = 9; H₁: σ² < 9

H_o: σ² = 9; H₁: σ² ≠ 9

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume an exponential population distribution.

We assume a binomial population distribution.    

We assume a normal population distribution.

We assume a uniform population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100   

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude that the new typhoid shot has a smaller variance of protection times.

At the 5% level of significance, there is sufficient evidence to conclude that the new typhoid shot has a smaller variance of protection times.    

(f) Find a 90% confidence interval for the population standard deviation. (Round your answers to two decimal places.)

lower limit
upper limit

Interpret the results in the context of the application.

We are 90% confident that σ lies above this interval.

We are 90% confident that σ lies within this interval.    

We are 90% confident that σ lies outside this interval.

We are 90% confident that σ lies below this interval.

Answer 1

a)
0.05

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: σ^2 = 9
Alternative Hypothesis, Ha: σ^2 < 9

Test statistic,
Χ^2 = (n-1)*s^2/σ^2
Χ^2 = (25 - 1)*2.25/9
Χ^2 = 6

DF = 24

normal population distribution

c)
P-value = 0.0001

P-value < 0.005

d)
option C

e)
At the 5% level of significance, there is sufficient evidence to conclude that the new typhoid shot has a smaller variance of protection times.

f)
Here s = 1.5 and n = 25
df = 25 - 1 = 24

α = 1 - 0.9 = 0.1
The critical values for α = 0.1 and df = 24 are Χ^2(1-α/2,n-1) = 13.848 and Χ^2(α/2,n-1) = 36.415

CI = (sqrt(24*1.5^2/36.415) , sqrt(24*1.5^2/13.848))
CI = (1.22 , 1.97)

We are 90% confident that σ lies below this interval.