Question

The number of ants per colony (in thousands) was determined for two species of ants. For the first species of ants, 10 colonies produced a mean of 17.22 and a standard deviation of 3.74. For the second species of ant, 15 colonies produced a mean of 15.18 and a standard deviation of 2.44. Assume that the number of ants per colony is normally distributed.

a.) Set up the null and alternative hypothesis needed to obtain the statistical evidence needed to support the theory that there is a significant difference between the mean number of ants for the two species of ant.

b.) Do you think that "pooling" of variances is appropriate in this case? Why or why not? Justify your answer.

c.) Based on your answer from part b, calculate the value of the test statistic for the hypothesis from part a, state the degrees of freedom, and find the approximate value of the p-value.  

d.) Would you reject H₀ or fail to reject H₀ at a 5% level of significance?

e.) What do you conclude about the mean amount of pollen for the two varieties of flowers?

Answer 1

MINITAB used for calculations

17.) The number of ants per colony (in thousands) was determined for two species of ants. For the first species of ants, 10 colonies produced a mean of 17.22 and a standard deviation of 3.74. For the second species of ant, 15 colonies produced a mean of 15.18 and a standard deviation of 2.44. Assume that the number of ants per colony is normally distributed.

a.) Set up the null and alternative hypothesis needed to obtain the statistical evidence needed to support the theory that there is a significant difference between the mean number of ants for the two species of ant.

Ho: µ1 = µ2   H1: µ1 ≠ µ2

b.) Do you think that "pooling" of variances is appropriate in this case? Why or why not? Justify your answer.

Test and CI for Two Variances

Method

σ₁: standard deviation of Sample 1

σ₂: standard deviation of Sample 2

Ratio: σ₁/σ₂

F method was used. This method is accurate for normal data only.

Descriptive Statistics

Sample	N	StDev	Variance	95% CI for σ
Sample 1	10	3.740	13.988	(2.573, 6.828)
Sample 2	15	2.440	5.954	(1.786, 3.848)

Ratio of Standard Deviations

Estimated Ratio	95% CI for Ratio using F
1.53279	(0.856, 2.987)

Test

Null hypothesis	H₀: σ₁ / σ₂ = 1
Alternative hypothesis	H₁: σ₁ / σ₂ ≠ 1
Significance level	α = 0.05

Method	Test Statistic	DF1	DF2	P-Value
F	2.35	9	14	0.147

Calculated F=2.35, P=0.147 which is > 0.05 level of significance. Ho is not rejected. we conclude that variances of the two groups are equal. We conclude that pooling of variances is appropriate in this case.

c.) Based on your answer from part b, calculate the value of the test statistic for the hypothesis from part a, state the degrees of freedom, and find the approximate value of the p-value.  

Two-Sample T-Test and CI

Method

μ₁: mean of Sample 1

µ₂: mean of Sample 2

Difference: μ₁ - µ₂

Equal variances are assumed for this analysis.

Descriptive Statistics

Sample	N	Mean	StDev	SE Mean
Sample 1	10	17.22	3.74	1.2
Sample 2	15	15.18	2.44	0.63

Estimation for Difference

Difference	Pooled StDev	95% CI for Difference
2.04	3.02	(-0.51, 4.59)

Test

Null hypothesis	H₀: μ₁ - µ₂ = 0
Alternative hypothesis	H₁: μ₁ - µ₂ ≠ 0

T-Value	DF	P-Value
1.66	23	0.111

Test statistic = 1.66 and P value =0.111

d.) Would you reject H0 or fail to reject H0 at a 5% level of significance?

Since obtained p value 0.111 is > 0.05 level of significance, we fail to reject H0.

e.) What do you conclude about the mean amount of pollen for the two varieties of flowers?

There is not sufficient evidence to conclude that there is a significant difference between the mean number of ants for the two species of ant.