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 23 people were given the new shot. Regular blood tests showed that the sample standard deviation of protection times was 2.1 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}*:

(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 uniform population distribution. We assume a exponential population distribution.We assume a binomial 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 < 0.010*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 outside this
interval. We are 90% confident that
*σ* lies within this interval.We are 90% confident that
*σ* lies below this interval.

Answer #1

a) level of significance =0.05

*H _{o}*:

b)

value of the chi-square statistic =(23-1)*(2.1/3)^{2}
=10.78

degrees of freedom =23-1=22

We assume a normal population distribution.

c) 0.010 < *P*-value < 0.025

d)

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

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)

lower limit =1.69

upper limit =2.80

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

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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 19 people were given the new shot. Regular blood tests
showed that the sample standard deviation of protection times was
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