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 21 people were given the new shot. Regular blood tests showed that the sample standard deviation of protection times was 1.3 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 > 9H_o: σ² = 9; H₁: σ² < 9    H_o: σ² < 9; H₁: σ² = 9H_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 uniform population distribution.We assume a exponential population distribution.    We assume a binomial population distribution.We assume a normal population distribution.

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

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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 within this interval.We are 90% confident that σ lies below this interval.    We are 90% confident that σ lies above this interval.We are 90% confident that σ lies outside this interval.

Answer 1

Answer:

(a) What is the level of significance?

Level of significance = α = 0.05

State the null and alternate hypotheses.

H_o: σ² = 9; H₁: σ² < 9

We are given σ = 3, so σ² = 9

(b) Find the value of the chi-square statistic for the sample.

We are given n = 21, S = 1.3

Chi square = (n – 1)*S^2/ σ²= (21 - 1)*1.3^2/9 = 3.7556

Chi square = 3.7556

What are the degrees of freedom?

df = n – 1 = 21 – 1 = 20

What assumptions are you making about the original distribution?

We assume a normal population distribution.

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

P-value = 0.0021

(by using Chi square table)

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 reject the null hypothesis.

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

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