Question

A researcher wants to test whether the mean lengths of two species of trout are the same. He obtains the weights of 10 individuals of species A and 10 individuals of species B. The sample mean weights are 12.3 kg and 14.3 kg, respectively, and the sample variances are 3.5 kg² and 4.5kg². Assume that the lengths are normally distributed.  

a. Should we retain the hypothesis that the population variances are equal?

b. Should we retain the hypothesis that the population means are equal?

Answer 1

Since the level of significance is not mentioned, we take the default value, = 0.05

(a) F TEST FOR VARIANCES:

The Hypothesis

H0:

Ha:

The Test Statistic: Since s2² is > s1²

F = s₂²/s₁² = 4.5/3.5 = 1.2857

The p Value at F = 1.2857, df1 = 9, df2 = 9 ; Using the excel function FDIST(1.2857,9,9) we get the right tailed p value, which is 0.3571. Therefore the 2 tailed p value = 2 * 0.3571 = 0.7142

The Decision Rule: If p value is < , Then reject H0.

The Decision: Since p value is > , we fail to reject H0.

Therefore Yes, we retain the hypothesis that the population variances are equal.

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(b) The 2 sample t test for means, assuming unequal variances.

The Hypothesis:

H0:

Ha:

The test statistic:

Since it is given that the population is normally distributed, but population standard deviation are unknown and sample sizes are < 30, we use the 2 sample t test.

The p value: The p value (2 tail) for t = -2.24 and degrees of freedom = 10 + 10 - 2 = 18 is;

p value = 0.0379

The Decision Rule: If p value is < , Then Reject H0.

The Decision: Since p value (0.0379) is < (0.05), we reject H0.

Therefore No, we do not retain the hypothesis that the population means are equal.

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