1) Modern
medical practice tells us not to encourage babies to become too fat. Is there a
positive correlation between the weightxof a
1-year old baby and the weightyof the mature adult (30 years old)? A random sample of
medical files produced the following information for 14 females:
|
x(lb) |
23 |
26 |
23 |
23 |
20 |
15 |
25 |
21 |
17 |
24 |
26 |
22 |
18 |
19 |
|
y(lb) |
120 |
125 |
124 |
122 |
130 |
120 |
145 |
130 |
130 |
130 |
130 |
140 |
110 |
115 |
Complete
parts (a) through (c), given?x=302,?y=1771,?x2=6664,?y2=225,135, and?xy=38,378.
(a)
Findx,y, andb. (Round
your answers forxandyto two decimal places. Round your answer forbto
four decimal places.)
|
x |
= |
|
|
y |
= |
|
|
b |
= |
b) Find the equation of the least-squares line.
(Round your answers to four decimal places.)
|
|
= |
+ x |
(c) Find the sample correlation coefficientrand
the coefficient of determination. (Round your answers to three decimal places.)
|
r |
= |
|
|
r2 |
= |
d) What percentage of variation in y is
explained by the least-squares model? (Round your answer to one decimal place.)
%
(e) If a female baby weighs16 pounds at 1 year, what do you
predict she will weigh at 30 years of age? (Round your answer to two decimal
places.)
lb
2) Suppose
you took random samples from three distinct age groups. Through a survey, you
determined how many respondents from each age group preferred to get news from
T.V., newspapers, the Internet, or another source (respondents could select
only one mode). What type of test would be appropriate to determine if there is
sufficient statistical evidence to claim that the proportions of each age group
preferring the different modes of obtaining news are not the same? Select from
tests of independence, homogeneity, goodness-of-fit, and ANOVA.
Since
we are determining if the current distribution of fits the previous
distribution of responses, the goodness-of-fit test is appropriate.
Since
we are comparing three distinct age groups, the test of two-way ANOVA is
appropriate.
Since
we are interested in proportions, the test for homogeneity is appropriate.
Since
we can claim all the variables are independent, the test of independence is
appropriate.
Since
we are comparing to a fixed variance, the test of ANOVA is appropriate.
3) Professor
Stone complains that student teacher ratings depend on the grade the student
receives. In other words, according to Professor Stone, a teacher who gives
good grades gets good ratings, and a teacher who gives bad grades gets bad
ratings. To test this claim, the Student Assembly took a random sample of 300
teacher ratings on which the student’s grade for the course also was indicated.
The results are given in the following table. Test the hypothesis that teacher
ratings and student grades are independent at the 0.01 level of significance.
|
Rating |
A |
B |
C |
F (or |
Row |
|
Excellent |
10 |
17 |
12 |
5 |
44 |
|
Average |
25 |
31 |
73 |
14 |
143 |
|
Poor |
21 |
28 |
44 |
20 |
113 |
|
Column Total |
56 |
76 |
129 |
39 |
300 |
(a) Give the value of the level of significance.
(b)
State
the null and alternate hypotheses.
H0:
Student grade and teacher rating are independent.
H1: Student grade and teacher rating are not independent.
H0:
The distributions for the different ratings are the same.
H1: The distributions for the different ratings are
different.
H0:
TestsA,B,C,F (or
withdrawal) are independent.
H1: TestsA,B,C,F (or
withdrawal) are not independent.
H0:
Ratings of excellent, average, and poor are independent.
H1: Ratings of excellent, average, and poor are not
independent.
(c)
Find the sample test statistic. (Round your
answer to two decimal places.)
d)
Find or estimate theP-value of the sample test
statistic. (Round your answer to three decimal places.)
e)Conclude the test.
SinceP-value ??,
we reject the null hypothesis.
SinceP-value
<?, we reject the null hypothesis.
SinceP-value ??,
we do not reject the null hypothesis.
SinceP-value
<?, we do not reject the null hypothesis.
4)
A
sociologist is studying the age of the population in Blue Valley. Ten years
ago, the population was such that21%were under 20 years old,11%were in the 20- to
35-year-old bracket,33%were between 36 and 50,24%were between 51 and 65, and11%were over 65. A study done
this year used a random sample of 210 residents. This sample is given below. At
the 0.01 level of significance, has the age distribution of the population of
Blue Valley changed?
|
Under 20 |
20 – 35 |
36 – 50 |
51 – 65 |
Over 65 |
|
29 |
25 |
68 |
67 |
21 |
a)
Give the value of the level of significance.
b)
State
the null and alternate hypotheses. (Please highlight the correct answer).
H0:
The population 10 years ago and the population today are independent.
H1: The population 10 years ago and the population today are
not independent.
H0:
Ages under 20 years old, 20- to 35-year-old, between 36 and 50, between 51 and
65, and over 65 are independent.
H1: Ages under 20 years old, 20- to 35-year-old, between 36
and 50, between 51 and 65, and over 65 are not
independent.
H0:
The distributions for the population 10 years ago and the population today are
the same.
H1: The distributions for the population 10 years ago and the
population today are different.
H0:
Time ten years ago and today are independent.
H1: Time ten years ago and today are not independent.
c)
Find the sample test statistic. (Round your
answer to two decimal places.)
d)
Find or estimate theP-value of the sample test
statistic. (Round your answer to three decimal places.)
e)
Conclude
the test.
SinceP-value
<?, we reject the null hypothesis.
SinceP-value ??,
we do not reject the null hypothesis.
SinceP-value ??,
we reject the null hypothesis.
SinceP-value
<?, we do not reject the null hypothesis.
