Question

A doctor released the results of clinical trials for a vaccine to prevent a particular disease. In these clinical​ trials, 400 comma 000 children were randomly divided in two groups. The subjects in group 1​ (the experimental​ group) were given the​ vaccine, while the subjects in group 2​ (the control​ group) were given a placebo. Of the 200 comma 000 children in the experimental​ group, 38 developed the disease. Of the 200 comma 000 children in the control​ group, 81 developed the disease. Does it appear to be the case that the vaccine was​ effective? Use the alphaequals0.01 level of significance.

Answer 1

To determine if the vaccine is effective, we conduct a hypothesis test using a significance level of 0.01. The proportion of children who developed the disease in the experimental group was 0.00019, while in the control group it was 0.000405. Calculating the test statistic, we find that it is -12.231, which is smaller than the critical value. Therefore, we can reject the null hypothesis and conclude that the vaccine is effective.

Step-by-step explanation:

To determine whether the vaccine is effective, we need to conduct a hypothesis test.

H0 (Null hypothesis): The vaccine has no effect.

HA (Alternative hypothesis): The vaccine is effective.

We will use a significance level (alpha) of 0.01.

First, calculate the proportions of children who developed the disease in each group. In the experimental group, 38 out of 200,000 children developed the disease, which is 0.00019. In the control group, 81 out of 200,000 children developed the disease, which is 0.000405.

Next, calculate the test statistic using the formula: z = (p1 - p2) / sqrt((p1(1-p1))/n1 + (p2(1-p2))/n2), where p1 and p2 are the proportions, and n1 and n2 are the sample sizes.

Substituting the values, we get z = (0.00019 - 0.000405) / sqrt((0.00019(1-0.00019))/200,000 + (0.000405(1-0.000405))/200,000) = -12.231.

Looking up the critical value for a significance level of 0.01 in a z-table, we find that it is -2.33.

Since the calculated test statistic (-12.231) is less than the critical value (-2.33), we can reject the null hypothesis. Therefore, it appears that the vaccine is effective.

Answer 2

We conclude that the vaccine appears to be​ effective.

Step-by-step explanation:

We are given that a doctor released the results of clinical trials for a vaccine to prevent a particular disease.

The subjects in group 1​ (the experimental​ group) were given the​ vaccine, while the subjects in group 2​ (the control​ group) were given a placebo. Of the 200 comma 000 children in the experimental​ group, 38 developed the disease. Of the 200 comma 000 children in the control​ group, 81 developed the disease.

Let
`p_1` = proportion of subjects in the experimental​ group who developed the disease.

`p_2` = proportion of subjects in the control​ group who developed the disease.

So, Null Hypothesis,
`H_0` :
`p_1\geq p_2` {means that the vaccine does not appears to be​ effective}

Alternate Hypothesis,
`H_A` :
`p_1<p_2` {means that the vaccine appears to be​ effective}

The test statistics that would be used here Two-sample z proportion statistics;

T.S. =
`\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{(\hat p_1(1-\hat p_1))/(n_1)+(\hat p_2(1-\hat p_2))/(n_2) } }` ~ N(0,1)

where,
`\hat p_1` = sample proportion of children in the experimental​ group who developed the disease =
`(38)/(200,000)` = 0.00019

`\hat p_2` = sample proportion of children in the control​ group who developed the disease =
`(81)/(200,000)` = 0.00041

`n_1` = sample of children in the experimental​ group = 200,000

`n_2` = sample of children in the control​ group = 200,000

So, test statistics =
`\frac{(0.00019-0.00041)-(0)}{\sqrt{(0.00019(1-0.00019))/(200,000)+(0.00041(1-0.00041))/(200,000) } }`

= -4.02

The value of z test statistics is -4.02.

Now, at 0.01 significance level the z table gives critical values of -2.33 for left-tailed test.

Since our test statistics is less than the critical value of z as -4.02 < -2.33, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the vaccine appears to be​ effective.