Question

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In a given year, the rate of flu infection for the general public was 8.3%. In a sample of 200 people who received the flu vaccine, the rate of flu infection was just 6.5%. What conclusion should you draw?

a. Do not reject the null hypothesis; flu rates for vaccinated people are lower
b. Reject the null hypothesis; flu rates for vaccinated people are not lower
c. Do not reject the null hypothesis; flu rates for vaccinated people are not lower
d. Reject the null hypothesis; flu rates for vaccinated people are lower

Answer 1

d. Reject the null hypothesis; flu rates for vaccinated people are lower

Explanation:

To determine the conclusion, we need to perform a hypothesis test.

The null hypothesis is that the flu rates for vaccinated people are the same as the flu rates for the general public. The alternative hypothesis is that the flu rates for vaccinated people are lower than the flu rates for the general public.

We can use a one-tailed z-test for proportions to test this hypothesis, where:

• p1 is the proportion of flu infections in the vaccinated sample (6.5% or 0.065)
• p2 is the proportion of flu infections in the general public (8.3% or 0.083)
• n is the sample size (200)

The test statistic is:

z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2))

where p is the pooled proportion (the weighted average of the two proportions):

p = (x1 + x2) / (n1 + n2)

and x1 and x2 are the number of flu infections in the vaccinated sample and the general public, respectively.

Using this formula, we get:

p = (0.065 x 200 + 0.083 x 200) / (200 + 200) = 0.074

z = (0.065 - 0.083) / sqrt(0.074 x 0.926 x (1/200 + 1/200)) = -1.87

The critical value for a one-tailed test at the 5% level of significance is -1.645. Since our calculated z-value (-1.87) is less than the critical value (-1.645), we reject the null hypothesis and conclude that the flu rates for vaccinated people are lower than the flu rates for the general public.

Therefore, the answer is (d) Reject the null hypothesis; flu rates for vaccinated people are lower.

Answer 2

c) Do not reject the null hypothesis; flu rates for vaccinated people are not lower.

Explanation:

In a given year, the rate of flu infection for the general public was 8.3%. In a sample of 200 people who received the flu vaccine, the rate of flu infection was just 6.5%.

To draw a conclusion, we need to perform a hypothesis test to determine whether the rate of flu infection for vaccinated people is significantly lower than the rate of flu infection for the general public.

The population parameter p is the proportion of the people who were infected by the flu.

If the number of people infected by the flu has not changed then the null hypothesis is:

H₀: p = 0.083

If the number of people infected by the flu has decreased then the alternative hypothesis is:

H₁: p < 0.083

The alternative hypothesis specifies that p is less than 0.083, so the test is one-tailed.

Let X be the number of people in the sample who were infected by the flu.

The sampling distribution of X under H₀ is binomial with p = 0.083 and 200 trials, so:

X ~ B(200, 0.083)

As we have not been given a significance level, assume the significance level is 5%. Therefore, α = 0.05.

If 6.5% of a sample of 200 people were infected by the flu, then 13 people out of the sample were infected by the flu.

The p-value is the probability of X being 13 or less, under the null hypothesis.

Using a calculator, P(X ≤ 13) = 0.217 (3 s.f.).

Since 0.217 > 0.05, the result is not significant.

Therefore, there is insufficient evidence at the 5% level of significance to reject H₀ in favour of the claim that the rate of flu infection for vaccinated people is significantly lower than that of the general public.