Stats 120B/Math 131B: Midterm Exam 2, Winter 2014

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Stats 120B/Math 131B: Midterm Exam 2, Winter 2014

Version A Problem 1/Version B Problem 3:

Version A:

AssumeX1; : : : ;X25

iid_N(1; _2).

(a) [5 pts] The MLE of_2is

^_2=

P25

i=1(Xi????1)2

25

:

Show that 25^_2=_2has a Chi-squared distribution with 25 degrees of freedom.

(b) [5 pts] Use the result in part (a) to derive a 99% con_dence interval for_2. (Your

quantiles should have numerical values { not just symbols.) Your _nal answer should

be in interval form (L;U), for some lower boundLand upper boundU.

Version B:AssumeX1; : : : ;X15

iid_N(1; _2).

(a) [5 pts] The MLE of_2is

^_2=

P15

i=1(Xi????1)2

15

:

Show that 15^_2=_2has a Chi-squared distribution with 15 degrees of freedom.

(b) [5 pts] Use the result in part (a) to derive a 90% con_dence interval for_2. (Your

quantiles should have numerical values { not just symbols.) Your _nal answer should

be in interval form (L;U), for some lower boundLand upper boundU.

Version A Problem 2/Version B Problem 4:

Version A:

Edgar Anderson collected data on sepal widths (in centimeters) of theSetosaspecies

of iris ower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random

sample of 18 owers, 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.

.jpg">

Using a significance level of .05, conduct a one-samplet-test to determine if the mean

sepal width in the population ofSetosaspecies of iris owers di_ers from 3.2 cm:

(a) [2 pts] State hypotheses in terms of the population parameter. (De_ne 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 _nd the p-value.

(d) [1 pt] State your decision (RejectH0or Fail to rejectH0).

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

Version B:

Edgar Anderson collected data on sepal widths (in centimeters) of theVirginica

species of iris ower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random

sample of 18 owers, 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.

.jpg">

Using a significance level of .05, conduct a one-samplet-test to determine if the mean

sepal width in the population ofVirginicaspecies of iris flowers differs from 2.9 cm:

(a) [2 pts] State hypotheses in terms of the population parameter. (De_ne 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 _nd the p-value.

(d) [1 pt] State your decision (RejectH0or Fail to rejectH0).

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

Version A Problem 3/Version B Problem 5:

Assume thatX1; : : : ;Xn

iid_Exp(1). De_ne the minimum of the sample as

Y= min(X1; : : : ;Xn):

Derive the sampling distribution ofY. (Hint:First derive the cdf ofY. Then use the

cdf to _nd the pdf. The pdf will match the form of one of the probability distributions

on our distribution summary table.)

Version A Problem 4/Version B Problem 1:

Version A:

De_ne the following random variables:

X1;X2;X3

iid_N(1;9); Z_N(0;1); V1__2(10); V2__2(20):

Assume all variables are independent of one another. Also de_ne _X

= (X1+X2+X3)=3.

(a) [2 pts] Use the variables de_ned above to de_ne a random variable that has anF

distribution. Specify both the numerator and the denominator degrees of freedom

for yourFdistribution.

(b) [2 pts] Use the variables de_ned above to de_ne a random variable that has a_2(1)

distribution.

(c) [3 pts] State the probability distribution (name and parameter values) of

Y= _X????1 3Z

:

Justify your answer.

(d) [3 pts] State the probability distribution (name and parameter values) of

W= _X

+ 5Z+ 2:

Justify your answer.

(c) [3 pts] State the probability distribution (name and parameter values) of

W=Z+ 2 _X

+ 5:

Justify your answer.

Version A Problem 5/Version B Problem 2:

Version A:

For a random sample of 20 women, aged 18 to 29, responses to the question \How

tall would you like to be?" were recorded along with the woman's actual height. Let

Xi= di_erence between the desired height and actual height (desired????actual) in inches

for theith randomly selected woman, and assumeXi

iid_N(_; _2),i= 1; : : : ;20, where

_and_2are unknown parameters.

(a) [1 pt] In words, what does the parameter_represent in the context of this study?

(b) [3 pts] In the observed sample, the mean difference between the desired height and

the actual height (desired????actual) was 1.8 inches. The standard deviation of the differences in the sample was 2.1 inches. Calculate a 95% confidence interval for_.

(c) [2 pts] Provide an interpretation of the interval in part (b)in context of the problem.

(If you are unable to calculate an interval, just make up numbers for the endpoints

and use your made-up interval for the remainder of this problem.)

(d) [2 pts] Explain what is meant by the phrase \95% confidence."

(e) [2 pts] Does your interval in part (b) provide signi_cant evidence that, on average,

women aged 18 to 29 desire to be taller than they actually are? Justify your answer.

Version B:

For a random sample of 25 women, aged 18 to 29, responses to the question \How

tall would you like to be?" were recorded along with the woman's actual height. Let

Xi= di_erence between the desired height and actual height (desired????actual) in inches

for theith randomly selected woman, and assumeXi

iid_N(_; _2),i= 1; : : : ;25, where

_and_2are unknown parameters.

(a) [1 pt] In words, what does the parameter_represent in the context of this study?

(b) [3 pts] In the observed sample, the mean di_erence between the desired height and

the actual height (desired????actual) was 2.3 inches. The standard deviation of the

di_erences in the sample was 3.0 inches. Calculate a 95% con_dence interval for_.

(c) [2 pts] Provide an interpretation of the interval in part (b)in context of the problem.

(If you are unable to calculate an interval, just make up numbers for the endpoints

and use your made-up interval for the remainder of this problem.)

(d) [2 pts] Explain what is meant by the phrase \95% con_dence."

(e) [2 pts] Does your interval in part (b) provide signi_cant evidence that, on average,

women aged 18 to 29 desire to be taller than they actually are? Justify your answer.

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