Unit_4_Review MY ANSWERS pdf
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Jan 9, 2024
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Elementary Statistics Unit 4 : Review Your solutions should be clear, complete, and sufficiently detailed in order to demonstrate your understanding and communicate your reasoning and method of solving the problem. Each problem will be evaluated on a 5-point standard rubric. Student’s Name Problem 1 The investment firm Brown & Johnson reported that 52.7% of 300 randomly selected American households owned stocks. Does this result provide sufficient sample evidence to indicate that a majority of households in America own stocks? Use α
= 0.05. Page 1 Mathematics 13 STATE HYPOTHESIS (MAJORITY = > 50%:
H0: p ≤ 50% H1: p > 50%
a = 0.05
x = n ⋅
pˆ = 300 ⋅
0.527 = 158.1 ≈ 158 (HAS TO BE DISCRETE)
P-VALUE = 0.178
SINCE 0.178 IS NOT 0.05 OR LESS WE DO NOT REJECT H0
Since, our decision is to not reject H0 this means that our result does not provide sufficient evidence to indicate that a majority of households in America own stocks.
Elementary Statistics Unit 4 : Review Problem 2 The results of a nationwide Gallup poll concerning a person’s race and their opinion on the death penalty are summarized in the table below. Person’s Race White Black Hispanic Person’s Opinion on
the Death
Penalty
In Favor 925 277 394 Opposed 362 415 317 (a) Is a person’s opinion on the death penalty independent of their race at the 1% level of significance? (b) Based on these results, what can one conclude about the relationship between a person’s race and their opinion on the death penalty? Page 2 Mathematics 13 STATE HYPOTHESIS:
H0 : The events are independent.
H1 : The events are not independent.
α = 1%
P-VALUE = 0
SINCE P-VALUE IS LESS THAN 0.001 WE
REJECT H0
Because of our decision to reject H0, at the 1% significance level, the persons opinoin on the death penalty is not independent of their race. Based on our earlier results, we can conclude that a persons opinon on the death penalty is in fact
related to their race. Our result table shows us that whites and hispanics are more likely to be in favor of the
death penalty while blacks are less likely to be in favor.
Elementary Statistics Unit 4 : Review Problem 3 The total stopping distance (in feet) was measured for a midsize four-door sedan driving in dry conditions at various speeds. The resulting data are presented in the table below. Speed (in mph) 10 20 30 40 50 60 70 80 Total Stopping Distance 27 61 104 168 235 297 386 473 (a) Determine the linear regression model that will best predict the total stopping distance for a midsize four-door sedan driving in dry conditions based on the speed of the vehicle. (b) How well does the linear regression model fit this sample data? (c) Predict the total stopping distance for a midsize four-door sedan driving at a speed of 65 mph in dry conditions. Page 3 Mathematics 13 The linear regression model is yˆ = 6.42x − 70.04
r2 = 0.981
Yes, the linear regression model fits the sample well because the coefficitent of determination
r2 = 0.981 is quite close to 1.
For x = 65, yˆ = 6.42 ⋅
(65) − 70.04 ≈ 347.26
This means, the total stopping distance for a midsize four-door sedan driving at a speed of 65 mph in dry conditions is predicted to be 347.3 feet.
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Elementary Statistics Unit 4 : Review Problem 4 Every day for an entire week, a cup of coffee was purchased at Starbucks, McDonalds, and Dunkin Donuts. The temperature (in °
F) was determined immediately after the cup of coffee was served. The following table contains the sample results. Establishment Serving Temperature (in °
F) of Cup of Coffee Starbucks 173 168 172 174 171 172 170 McDonalds 179 181 180 182 177 184 178 Dunkin Donuts 176 168 164 170 172 169 165 Conduct a hypothesis test using α
= 0.05 to determine whether the mean serving temperature of a cup of coffee is the same at Starbucks, McDonalds, and Dunkin Donuts. Page 4 Mathematics 13 STATE HYPOTHESIS:
H0 : All of the population means are equal. H1 : Not all of the population means are equal.
Use α = 0.05
P-VALUE=0
SINCE THE P-VALUE IS LESS THAN 0.05 OUR DECISION IS TO REJECT H0
This means, that the mean serving temperature of a cup of coffee at Starbucks, Mcdonalds, and Dunkin Donuts
is not all the same.
Elementary Statistics Unit 4 : Review Problem 5 Use the sample data collected in Problem 4 to construct all pairwise comparison confidence intervals to estimate the difference in the mean serving temperature of a cup of coffee at Starbucks, McDonalds, and Dunkin Donuts with a simultaneous confidence level of 95%. Interpret the results. Page 5 Mathematics 13 3 pairwise comparison confidence intervals individual confidence level = 1 -
α = 3 0.95 ≈ 0.983
−12 < μ1 − μ2 < −5.4
−2.8 < μ1 − μ3 < 7.4
5.8 <
μ2 − μ3 < 16.2
These results mean, the mean serving temperature of a cup of coffee at McDonalds is anywhere from 5.4°F to 12°F higher than that at Starbucks and anywhere from 5.8°F to 16.2°F higher than that at Dunkin Donuts. We can also conclude that their is no significant difference in the mean coffee serving temperature at Starbucks and Dunkin.
Elementary Statistics Unit 4 : Review Problem 6 The Fish and Game Department stocked Lake Loomis with fish according to the following percentages: 10% catfish, 30% bass, 40% bluegill, and 20% pike. Five years later they sampled the lake to see if the distribution of fish had changed. Of the 500 fish in their sample, 62 were catfish, 108 were bass, 204 were bluegill, and 126 were pike. Use a Chi-Squared test to determine whether the distribution of fish in Lake Loomis had changed significantly over this five-year period. Based on this result, what conclusion should the Fish and Game Department reach? Page 6 Mathematics 13 STATE HYPOTHESIS:
H0 : All of the expected frequencies fit the observed frequencies.
H1 : Not all of the expected frequencies fit the observed frequencies.
Use α = 0.05
n = 62 + 108 + 204 + 126 = 500
Ei = n ⋅
pi df = c − 1 = 4 − 1 = 3
P-VALUE = 0
SINCE OUR P-VALUE IS LESS THAN 0.05 OUR DECISION IS TO REJECT H0
Based on our decision the reject H0. not all of the expected frequencies fit the observed frequencies. So, based on this result, the Fish and Game Department should reach the conclusion that the distribution of fish in Lake Loomis had changed significantly over this five-year period. Our data specifically indicates that the proportion of Bass has decreased while the proportion of Pike has increased.
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