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 29 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 a normal population distribution.

We assume a exponential population distribution.   

We assume a uniform population distribution.

We assume a binomial 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 outside this interval.

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 below this interval.

Answer 1

Answer a

The level of significance is defined as the probability of rejecting a null hypothesis by the test when it is really true, which is denoted as α. That is, P (Type I error) = α.

The null and alternate hypotheses: H_o: σ² = 9; H₁: σ² < 9

Answer b

Test statistics for the test is

where N is the sample size and s is the sample standard deviation.

The more this ratio(s/σ₀ ) deviates from 1, the more likely we are to reject the null hypothesis.

We have N=29 ,s=1.5 and σ₀ =3

Therefor,

Critical Region: Reject the null hypothesis if   T< ²_α,N−1

where ²_.,N−1 is the critical value of the chi-square distribution with N - 1 degrees of freedom.

We assume a normal population distribution

Answer c

P-value of the sample test statistic is 0.00003

P-value < 0.005

Answer d

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

Answer 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.    

Answer f

90% confidence interval is given as

Lower limit =

Upper limit =

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