Question

The following data represent petal lengths (in cm) for independent random samples of two species of Iris. Petal length (in cm) of Iris virginica: x1; n1 = 35 5.1 5.6 6.2 6.1 5.1 5.5 5.3 5.5 6.9 5.0 4.9 6.0 4.8 6.1 5.6 5.1 5.6 4.8 5.4 5.1 5.1 5.9 5.2 5.7 5.4 4.5 6.4 5.3 5.5 6.7 5.7 4.9 4.8 5.7 5.1 Petal length (in cm) of Iris setosa: x2; n2 = 38 1.6 1.9 1.4 1.5 1.5 1.6 1.4 1.1 1.2 1.4 1.7 1.0 1.7 1.9 1.6 1.4 1.5 1.4 1.2 1.3 1.5 1.3 1.6 1.9 1.4 1.6 1.5 1.4 1.6 1.2 1.9 1.5 1.6 1.4 1.3 1.7 1.5 1.5 (a) Use a calculator with mean and standard deviation keys to calculate x1, s1, x2, and s2. (Round your answers to two decimal places.) x1 = s1 = x2 = s2 = (b) Let μ1 be the population mean for x1 and let μ2 be the population mean for x2. Find a 99% confidence interval for μ1 − μ2. (Round your answers to two decimal places.) lower limit upper limit (c) Explain what the confidence interval means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, is the population mean petal length of Iris virginica longer than that of Iris setosa? Because the interval contains only positive numbers, we can say that the mean petal length of Iris virginica is longer. Because the interval contains only negative numbers, we can say that the mean petal length of Iris virginica is shorter. Because the interval contains both positive and negative numbers, we cannot say that the mean petal length of Iris virginica is longer. (d) Which distribution did you use? Why? The Student's t-distribution was used because σ1 and σ2 are unknown. The standard normal distribution was used because σ1 and σ2 are known. The standard normal distribution was used because σ1 and σ2 are unknown. The Student's t-distribution was used because σ1 and σ2 are known. Do you need information about the petal length distributions? Explain. Both samples are large, so information about the distributions is needed. Both samples are large, so information about the distributions is not needed. Both samples are small, so information about the distributions is needed. Both samples are small, so information about the distributions is not needed.

Answer 1

(a)

x1 =5.47

s1 =0.56

x2 =1.49

s2 =0.21

(b)

Sample mean difference =x1 - x2 =5.47 - 1.49 =3.98

Standard Error, SE = =0.1004

For a two-tailed case, at 99% confidence level (1% significance level), at df =n1+n2-2 =35+38-2 =71, t-critical =2.6469

Margin of Error, MoE =t-critical*SE =2.6469*0.1004 =0.2657

99% confidence interval for μ1 − μ2 =(x1 - x2) MoE =3.98 0.2657 =(3.98 - 0.2657, 3.98+0.2657) =[3.71, 4.25].

Lower limit =3.71

Upper limit =4.25

(c)

Because the interval contains only positive numbers, we can say that the mean petal length of Iris virginica is longer.

(d)

The Student's t-distribution was used because σ1 and σ2 are unknown.

Both samples are large, so information about the distributions is not needed.

[n1 =35 > 30 (large sample) and n2 =38 > 30 (large sample)].