1.7
.7
.0
5. Cattl .
a)
~ot
b)
ot correct. A confidence interval i for the mean veight gain of the population of all
cow ,not indi idual cows.
c)
at carr ct. We don't n d a confidenc int TVal about the a rage weight gain for cOl~
in lhi lud. W' are certain that the mean weight gain of the cow in thi tud i 56
pounds. C nfiden e intervals are for th m an weight gain of the opulation of all cows.
correct. A confidence interval i for the mean weight gain of the population of all
cow . It says nothing about individual cow . This interpretation also app ars to imply
Ulal there b something 'pedal about the int rval that was g nerated, when this interval
actually one of many that could have been generated, d pending on the cow that were
chosen for the sample.
d) Not correct. This statement impli s that th average weight gain vaIie', It doesn't. We
just don't know w at it is, and we ar trying to find it. The average weight gain is either
between 45 and 67 pounds, or it isn't.
e)
at correct. This statem nt implies that thPTP ie:; something pecial about our interval,
when this int rval is acluall on uf many th t could haY been gen rated, depending
the cows that were chosen f r th sample. The correct interpr tarion is that 95 % of s
of thi size will produce an interval that will contain the mean" ight gain of the
population of all cows.
9. Pulse rates.
a) We are 95% confident the interval 70.9 to 74.5 beat p r
heart rate,
. ute contains the true mean
b) The width of the interval is ab ut 74.5 - 70.9 = 3.6 bats per minute. The margin of error is
half of that, about 1. beat per minute.
c) The margin of error would have been larger. More canfidenc r quire a larger critical
value of I, which increa e the margin of error.
15. Hot dogs.
a)
y ± <-1 (.};) = 310 ± r;9(
1:0)~
(298.5, 321.5)
b) W have a umed that the hot dog wight are independent and that the distribution
population of hot dog weight is ormal. The conditions to check are:
Randomization condition: We don't know that th hot dogs were sampl d at random
it i rea onable to think that the hot dogs are repre entative of hot dogs of thi type.
10% condition: 40 hot dogs are less than 10% of all hot dogs.
Nearly Normal condition: We don't have th actual data, but ince the sample of 40 h
dogs is fairly large, it is okay to proceed.
c) W are 95% confid nt that the interval 298.5 to 321.5 mg contains the true mean octiu
content in thi type of "reduced odium" hot dogs.
17. Second dog.
a) This larger sample should produc a more accurate estimate of the mean sodium con
the hot dogs. A larger random sample has a smaller tandard error, which results in
smaller margin of error.
b)
ECy)
=( In )=( ~)=
4.1 mg odium.
c) Randomization condition: We don't know that the hot dogs were ampled at rando
it is reasonable to think that the hot dogs are repr sentative of hot dogs of this type.
10% condition: 60 hot dogs are Ie than 10% of all hot dog .
Nearly Normal condition: We don't have the actual data, but since the ample of 60
dogs is large it is okay to proc ed.
The hot dogs in the sample had a mean sodium content of 318 mg, and a standard
deviation in odium content of 32 mg. Since the conditions have been atisfied, cons
one- ample I-interval, with 60 -1 = 59 degrees of freedom, at 95% confidence.
Y±l:_l(~)= 318±1;{~ )=C309.7, 326.3)
We are 95% confident that the interval 309.7 to 326.3 mg contains th
content of this type of hot dog.
d) If a "reduced sodium" hot dog ha to have at least 30% les odium than a hot dog
containing an average of 465 mg of sodium, then it must have less than 0.70(465) =3
mg of sodium. Since our 95% confidence interval extends above this value, there i n
enough evidence to suggest that this type of hot dog can be labeled as "reduced odi
19. Pin•.
If in fact the mean cholesterol of pizza eaters does not indicate a health risk, then only 7 out
of every 100 samples would be expected to have mean cholesterol as high or higher than
the mean cholesterol observed in the sample.
21. TV Safety.
a) The inspectors are performing an upper-tail test. They need to prove that the stands will
support 500 pounds (or more) easily.
b) The inspectors commit a Type I error if they certify the stands as safe, when they are not.
c) The inspectors commit a Type n error if they decide the stands are not safe, when they are.
23. TV safety revisited.
a) The value of IX should be decreased. This means a smaller chance of declaring the stands
safe under the null hypothesis that they are not safe.
b) The power of the test is the probability of correctly detecting that the stands can safely h
over 500 pounds.
c) 1) The company could redesign the stands so that their strength is mOre consistent, as
measured by the standard deviation. Redesigning the manufacturing process is likely 10
quite costly.
2) The company could increase the number of stands tested. This costs them both time
perform the test and money to pay the quality control personnel.
3) The company could increase IX, effectively lowering their standards for what is
required to certify the stands "safe". This is a big risk, since there is a greater chance of
Type I error, namely allowing unsafe tands to be sold.
4) The company could make the stands stronger, increasing the mean amount of weight
that the stands can safely hold. This type of redesign is expensive.
25. Marriage.
a) Ho: The mean age at which American m n first marry is 23.3 years. (Jl = 23.3)
H A : The mean age at which American men first marr 'i gr ater than 23.3 years. (fl >
b) Randomization condition: The 40 men were elected randomly.
10% condition: 40 men are less than 10% of all recently married men.
early ormal condition: The population of ages of men at fir t marriage is likely to be
kewed to the right. It is much more likely tllat ther are men who marry for the first time
at an older age than at an age that is very young. We should examine the distribution of
the sampl to check for eriou skewness and outliers, but with a large sample of 40 men, it
should be safe to proceed.
c) Since the condition for inference are satisfied, we can model the sampling distribution of
the mean age of men at first marriage with
N( 23.3, :In). Since we do not know a, the
standard deviation of the population, a(y) will be e timated by SE(-) =~, and we will
11
use a Student'
f
model, with 40 - 1 = 39 degree of freedom,
d) The mean ag at first marriage in
the sample was 24.2 years, with a
tandard d viation in age of 5.3
years. U e a one- ample He t,
mod ling the sampling
di tribution of y with
t39 ( 23.3,
~ )-
[3{ 23.3, J~o ).
Y- J1.
t=-S£(y)
f
=
24.2-23.3
5.3
140
t
- - - '39
= 1.07
The P-value is 0.1447.
e) If the mean age at first marriage is still 23.3 years, there is a 14.5% chance of getting a
sample mean of 24.2 years or older Simply from natural sampling variation.
f) Since the P-valu = 0.1447 is high, we fail to rej ct the null hypothesis. There is no evidence
to suggest that th mean age of men at first marriage ha changed from 23.3 years, the
mean in 1960.
27. Ruffles.
a) Randomization condition: The 6 bags were not elected at
random, but it is reasonable to thIDk that these bag are
representative of all bags of chips.
10% condition: 6 bag are less than 10% of all bags of chips.
early Normal condition: Tile histogram of the weight of chips in
the sample is nearly normal.
b) - = 28.78 grams
2
28.0
'" 0.40 gram
29.0
weight
C) Since the conditions for inference have been atisfied, use a
6 - 1 = 5 degrees of freedom, at 95 Yo confidence.
y±
3
one-~ample
(-lnLen'al,
ith
t:- (.In)= 28.78 =t;( 0.:0) = (28.36, 29.21)
1
d) We are 95% confident that th mean weight of the contents of Ruffle bags i::; b tween
and 29.21 grams.
e) Since the interval is above the stated weight of 28.3 grams, Lhere is evidence that the
company is filling the bags to more than the stat d weight, on average.
31. Cars.
Ho: The mean weight of cars currently licensed in the U.s. is 3000 pounds. (f.l = 3000)
HA: The mean weight of cars currently licensed in the U.S. is not 3000 pounds.
(Ji f-
Randomization condition: The 91 cars in the sample were randomly selected.
10% condition: 91 cars are less than 10% of all cars.
Nearly Normal condition: We don't have the actual data, so we cannot look at a gra
display, but since the sample is large, it is safe to proceed.
The cars in the sample had a mean weight of 2919 pounds and a standard deviation in
weight of 531.5 pounds. Since the conditions for inference are satisfied, we can model the
sampling di tribution of the mean weight of cars currently licensed in the U.s. with a
Student's r model, with 91 - 1
=
90 degrees of freedom, (90(3000, 53~~5).
We will perform a one-sample I-test.
ince the P-value = 0.1495 is
high, we fail to reject the null
hypoth sis. There is little
evidence to uggest that the
mean weight of cars in the U.S.
i different than 3000 pounds.
Y- P.
SE(y)
2919-3000
1=
531.5
91
1
. . - - - - p = 0.1495-----,
1=--
= -1.45
- ....IIIi!~:-----;;3.,dOOO=-----=~
...-
190
12. Portable phones.
Ho: The mean range of this type of portable phone is 150 feet. (/1 = 150)
HA: The mean range of this type of portable phone is not 150 feet. (/1 '" 150)
Randomization condition: The 44 phones in the sample were randomly selected.
10% condition: 44 phones are less than 10% of all cars.
Nearly Normal condition: We don't have the actual data, so we cannot look at a graphical
display, but since the sample is fairly large, it is safe to proceed.
The phones in the sample had a mean range of 142 feet and a standard deviation of 12 feet.
Since the conditions for inference are satisfied, we can model the sampling distribution of
the mean range of this type of portable phone with a Student's I model, with
44 - 1 = 43 degrees of freedom, 1<>(150, ~). We will perform a one-sample I-test.
Since the P-val ue = 0.0000654 is
very low, we reject the null
hypothesis. There is strong
evidence that the mean range
of this type of phone is not 150
feet. Our evidence suggests
Ihat the mean range is actually
less than 150 feet.
y - Po
f=--
SEC}')
112 - ]50
12
-!44
1'"
-4.42
[7fS;l
]42
f =--4.42
150
158
f
=4.42