HW 4 - homework assignment PDF

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HP340: Statistical Methods, Spring 2020 Due Date: February 25, 2020 Homework #4 Instructions: Clearly circle the letter of the best answer for multiple-choice questions and show your work when calculations are required. Where necessary, please explain your reasoning for the approach you used. 1. Consider the following population of 7 scores: Subject

1

2

3

4

5

6

7

Score

6

3

9

4

6

8

2

(This is a toy example created to help you cement the concept of sampling distribution of the mean. Of course the population mean of its 7 scores can be easily computed exactly and without any need for sampling to estimate it, but we will pretend otherwise for the sake of learning. In real situations, sampling is required because it would be too costly/difficult to examine each member of the very large populations we are typically interested in) a. List all possible different samples of 3 subjects (n = 3) that can be obtained from this population (you should get 35 samples in total) and compute the sample mean for each sample. The distribution of the 35 sample means is the ‘sampling distribution of the mean’ with n = 3 based on the ‘parent’ population given by the table above. (3 points)  At bottom! b. Construct a frequency distribution table of the sample distribution of the mean computed in part a. (3 points)  Using ranges of 1 Sample mean

Tally

Frequency

Rf

%f

Cf

Crf

C%f

7-7.99

IIII

4

0.1143

11.43

35

1.0

100

6-6.99

IIIIIIIII

9

0.2571

25.71

31

0.8857

88.57

5-5.99

IIIIIIIIIII

11

0.3143

31.43

22

0.6286

62.86

4-4.99

IIIIIIII

8

0.2286

22.86

11

0.3143

31.43

3-3.99

III

3

0.0857

8.57

3

0.0857

8.57

0

0

0

0

0

0

2-2.99

c.

Compute the population mean of the ‘parent’ population (the scores in the table above) and the mean of the sampling distribution of the mean. Compare the two and interpret. (3 points) o population mean = (6+3+9+4+6+8+2)/7 = 5.43

sampling distribution mean = (added sample means)/35 = 5.43  both are the same (rounding was used so might be slightly different) given the same sample was used and therefore the scores found and used to determine mean were all the same d. Graph a histogram of the sampling distribution of the mean. Comment on the shape of the distribution. Does it look normal? Would you expect it to look normal? Explain. (3 points)  The histogram is a normal bell-shaped curve meaning that the data found is normally distributed having a mean between 5-6 which is expected with a randomly selected sample of scores. (histogram at bottom!) o

2. A statistical test with a rejection region comprised of both tails of the sampling distribution of the test statistic is called a (an) _________ test. (Choose all that apply) (2 points) a. Two-tailed b. One-tailed c. Non-directional d. Directional e. None of the above 3.

When performing a two-sided z-test you obtain zobs = 2.10. If the significance level is set at  = 0.05, then the decision should be ______ the null hypothesis and _______ the alternative hypothesis. (2 points) a. Reject; do not reject b. Reject; accept c. Fail to reject; do not accept d. Fail to reject; accept e. Reject; reject

4.

Suppose a non-directional z-test was performed and the null hypothesis was rejected at a 5% significance level. Which of the following could have been the value of zobs in the statistical test? (Choose all that apply) (2 points) f. -1.98 g. -1.23 h. 0 i. D.1.69 j. None of the above

5.

Which of the following could be an appropriate null hypothesis for a statistical test concerning the mean cost of regular gas in LA county? (Choose all that apply.) (2 points) a.

H0 :

´ X ´ X

= $2.70/Gallon

H0 : ≠ $2.70/Gallon b. H 0 : μ = $2.70/Gallon c. d.

H 0 : μ ≠ $2.70/Gallon H

0 : μ ≥ $2.70/Gallon e. 6. Given the following information calculate the appropriate test statistics (zobs or tobs) for testing the equality of the population mean to the given value 0 and determine whether the test statistic falls into the appropriate two-tailed rejection region at the 0.05 significance level. (10 points)



´

μ0 =100, X = 25; X = 70, N=86  Z=70-100/(25/root86)= -11.13 o Test statistic=11.13 o Critical values: +/-1.96  11.13>1.96 so we reject the null hypothesis given there is significant information to decide that μ=/=100. ´ = 92, N=37 b. μ0 = 87, s ´X = 2.9; X  t=92-87/(2.9/root37)= 10.49 o df=36, alpha/2=0.25, critical values: +/- 2.028  10.47>2.028 so we so we reject the null hypothesis given there is significant information to decide that μ=/=87. a.

7.

You are analyzing a dataset of current rental prices from n = 97 randomly selected two-bedroom rental houses in a particular neighborhood of LA (neighborhood A). The average rent of the 97 houses is $1,832/month.

You want to compare the mean rental price of two-bedroom houses in this neighborhood to the mean rental price of 2-bedroom houses in a comparable neighborhood (neighborhood B) for which the current mean rental price of two-bedroom houses is known to be $1,765/month with a standard deviation of $234/month.(25 points) a. Identify the population and parameter of interest. (2 points)  The population is two-bedroom rental houses with the parameter of interest is the mean current rental prices b. What type of variable is rental price? (1 point)  Rental price is an independent variable c.

State the null an alternative hypothesis. 

(2 points)

Ho: μ =$1,832/month, Ha: μ =/=$1,832/month

You first assume that the standard deviation of rental prices in neighborhood A is the same as in neighborhood B: d. Perform a test of the null hypothesis you stated in part c at the 5% significance level and state your conclusion. (5 points) A=0.05, mean=1765, standard deviation=234, 95% CI At bottom! e. Compute a 95% CI for the parameter of interest. Explain how you could also perform the test in part d based on this CI. (5 points) At bottom! f. Extra credit: What is the largest significance level at which you don’t reject the null hypothesis? (2 points) 

99%

You then realize that the rental prices in neighborhood A seem to be more variable than those in neighborhood B, with an estimated standard deviation of $337. g. Perform a test of the null hypothesis in part c at the 5% significance level using now the estimated SD and state your conclusion. Contrast with part d. (5 points) At bottom! h. Compute a 95% CI for the parameter of interest using now the estimated standard deviation. Contrast with part e. (5 points) At bottom!...