The following data represent petal lengths (in cm) for independent random samples of two species of Iris.
Petal length (in cm) of Iris virginica: x_{1}; n_{1} = 35
5.3 | 5.9 | 6.5 | 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 | 5.2 |
Petal length (in cm) of Iris setosa: x_{2}; n_{2} = 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 | 1.6 |
(a) Use a calculator with mean and standard deviation keys to calculate x_{1}, s_{1}, x_{2}, and s_{2}. (Round your answers to two decimal places.)
x_{1} = | |
s_{1} = | |
x_{2} = | |
s_{2} = |
(b) Let μ_{1} be the population mean for
x_{1} and let μ_{2} be the
population mean for x_{2}. Find a 99% confidence
interval for μ_{1} − μ_{2}.
(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 standard normal distribution was used because σ_{1} and σ_{2} are unknown.The Student's t-distribution was used because σ_{1} and σ_{2} are unknown.
The Student's t-distribution was used because σ_{1} and σ_{2} are known.The standard normal distribution was used because σ_{1} and σ_{2} are known.
Do you need information about the petal length distributions?
Explain.
Both samples are small, so information about the distributions is not 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 large, so information about the distributions is needed.
a) x1= 5.50
s1=0.57
x2 =1.49
s2 =0.22
b)
std error =√(S^{2}_{1}/n_{1}+S^{2}_{2}/n_{2})= | 0.102 |
Point estimate of differnce =x1-x2 = | 4.005 | ||
for 99 % CI & 34 df value of t= | 2.728 | ||
margin of error E=t*std error = | 0.280 | ||
lower bound=mean difference-E = | 3.73 | ||
Upper bound=mean differnce +E = | 4.28 |
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 σ_{1} and σ_{2} are unknown.
e)
.Both samples are large, so information about the distributions is not needed
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