Question

to find out whether a new serum will arrest​ leukemia, ​mice, all with an advanced stage of the​ disease, are selected. mice receive the treatment and do not. survival​ times, in​ years, from the time the experiment commenced are provided. at the level of​ significance, can the serum be said to be​ effective? assume the populations to be normally distributed with equal variances.

Answer 1

No, the serum is not effective because the p-value for the hypothesis test is greater than the level of significance, and the null hypothesis that the means of the two populations are equal is not rejected.

We are given the hypothesis as:

H₀: μ₁ − μ₂ = 0

H₁: μ₁ − μ₂ ≠ 0 where,

H₀ = the null hypothesis,

H₁ = the alternative hypothesis.

Calculate the test statistic. The equation is:

t = x₁ - x₂ / √pooled variance(1/n₁ + 1/n₂)

Substituting the values, we get that:

t = 2.86 - 2.075 / 2.81(1/5 + 1/4) = 0.699

We need to find the value of 0.699 from the t lookup table and the two-tailed probability is 0.5071.

Therefore, No, the serum is not effective because the p-value for the hypothesis test is greater than the level of significance, and the null hypothesis that the means of the two populations are equal is not rejected.

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Answer 2

i) No, the serum is not effective because the p-value for the hypothesis test is greater than the level of significance, and the null hypothesis that the means of the two populations are equal is not rejected. ii) Yes, the assumption is justified. The Fstat was less than the Fcrit. therefore the null hypothesis that the variances of the two data sets are equal is not rejected.

Given: We know the hypothesis is as follows:

H₀: μ₁ − μ₂ = 0

H₁: μ₁ − μ₂ ≠ 0 where,

H₀ = the null hypothesis,

H₁ = the alternative hypothesis.

i) The table attached below shows the calculations of each data set's mean and standard deviation.

First: Calculate the pooled variance. The equation is:

$S^{2}_{p} = \frac{(n_{1} -1 )S^{2}_{1} + (n_{2} -1 )S^{2}_{2}}{(n_{1} -1 ) + (n_{2} -1 )}$ where,

$S^{2}_{p}$ is the pooled variance

n₁ = the sample size of the 1st sample = 5,

n₂ = the sample size of the 2nd sample = 4,

$S^{2}_{1}$ = the variance of sample 1

The standard deviation squared = 1.9705²,

$S^{2}_{2}$ is the variance of sample 2,

The standard deviation squared = 1.1673²

Now, substituting values:

$S^{2}_{p}$ = $\frac{(4 * 1.9705^2)+(3 * 1.1673^2)}{7}$ = 2.8027

Next, calculate the test statistic. The equation is:

tstat = $\frac{x_1 - x_2}{\sqrt{S^{2}_{p}}(\frac{1}{n_1} + \frac{1}{n_2} ) }$

Substituting values:

tstat = $\frac{2.86 - 2.075}{2.8027(\frac{1}{5} + \frac{1}{4} ) }$ = 0.699

Now, we need to find the value of 0.699 from the t lookup table and the two-tailed probability is 0.5071

ii) Yes, the assumption that the variances are equal is justified. The Fstat was less than the Fcrit.

Therefore, the null hypothesis that the variances of the two data sets are equal is not rejected.

The test statistic is  

Fstat = $\frac{S^{2}_{1}}{S^{2}_{2}}$

= $\frac{1.19705^2}{1.1673^2}$

= 1.0516

Now, from an F lookup table, we find the critical value is 15.1010.

As the test statistic is less than the critical value, the variances of the two samples are considered equal. Therefore, a pooled t-test is appropriate.

Hence:

i) No, the serum is not effective because the p-value for the hypothesis test is greater than the level of significance, and the null hypothesis that the means of the two populations are equal is not rejected.

(ii) Yes, the assumption is justified. The Fstat was less than the Fcrit. therefore the null hypothesis that the variances of the two data sets are equal is not rejected.

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Disclaimer: The given question is incomplete. The complete question is given below:

To find out whether a new serum will arrest leukemia, 9 mice, which have all reached an advanced stage of the disease, are selected. Five mice receive the treatment and 4 do not. The survival times, in years, from the time the experiment commenced are as follows:

Assume that the two distributions are normally distributed.

(i) Can the serum be said to be effective? Assume the variances are equal. Test at the 5% level.  

(ii) Is the assumption of equal variances made in (i) justified?