Stats 120B/Math 131B: Midterm Exam 2

Version A Problem 1/Version B Problem 3:
Version A: Assume X1,...,X25 iid ∼ N(1,σ2). (a) [5 pts] The MLE of σ2 is
ˆ σ2 =P25 i=1(Xi −1)2 25
.
Show that 25ˆ σ2/σ2 has a Chi-squared distribution with 25 degrees of freedom.

(b) [5 pts] Use the result in part (a) to derive a 99% confidence interval for σ2. (Your quantiles should have numerical values – not just symbols.) Your final answer should be in interval form (L,U), for some lower bound L and upper bound U.

Version B: Assume X1,...,X15 iid ∼ N(1,σ2). (a) [5 pts] The MLE of σ2 is
ˆ σ2 =P15 i=1(Xi −1)2 15
.
Show that 15ˆ σ2/σ2 has a Chi-squared distribution with 15 degrees of freedom.

(b) [5 pts] Use the result in part (a) to derive a 90% confidence interval for σ2. (Your quantiles should have numerical values – not just symbols.) Your final answer should be in interval form (L,U), for some lower bound L and upper bound U.

Version A Problem 2/Version B Problem 4:
Version A: Edgar Anderson collected data on sepal widths (in centimeters) of the Setosa species of iris flower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random sample of 18 flowers, the sample mean sepal width was 3.228 cm with a sample standard deviation of 0.329. A normal quantile-quantile plot of the sample widths is below.
-2 -1 0 1 2
2.42.62.83.03.23.43.6
Normal Q-Q Plot
Theoretical Quantiles
Sample Quantiles
Using a significance level of .05, conduct a one-sample t-test to determine if the mean sepal width in the population of Setosa species of iris flowers differs from 3.2 cm:
(a) [2 pts] State hypotheses in terms of the population parameter. (Define any symbols used.)

(b) [3 pts] Check assumptions and calculate the test statistic.

(c) [2 pts] Find the p-value. State which distribution table you used and where you looked on the table to find the p-value.

(d) [1 pt] State your decision (Reject H0 or Fail to reject H0).

(e) [2 pts] Provide a conclusion in terms of the research study.

Version B: Edgar Anderson collected data on sepal widths (in centimeters) of the Virginica species of iris flower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random sample of 18 flowers, the sample mean sepal width was 2.994 cm with a sample standard deviation of 0.372. A normal quantile-quantile plot of the sample widths is below.
-2 -1 0 1 2
2.62.83.03.23.43.63.8
Normal Q-Q Plot
Theoretical Quantiles
Sample Quantiles
Using a significance level of .05, conduct a one-sample t-test to determine if the mean sepal width in the population of Virginica species of iris flowers differs from 2.9 cm:
4
(a) [2 pts] State hypotheses in terms of the population parameter. (Define any symbols used.)

(b) [3 pts] Check assumptions and calculate the test statistic.

(c) [2 pts] Find the p-value. State which distribution table you used and where you looked on the table to find the p-value.

(d) [1 pt] State your decision (Reject H0 or Fail to reject H0).

(e) [2 pts] Provide a conclusion in terms of the research study.