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Unformatted text preview: Assignment 7 SOLUTIONS Chapter 19: # 8, 12, 18, 24, 26
Chapter 23: # 6, 12, 16, 17, 18 8. Confidence intervals, again. a) True. The smaller the margin of error is, the less confidence we have in the ability of our
interval to catch the population proportion. b) True. Larger samples are less variable, which translates to a smaller margin of error. We
can be more precise at the same level of confidence. c) True. Smaller samples are more variable, leading us to be less confident in the ability of
our interval to catch the true population proportion. d) True. The margin of error decreases as the square root of the sample size increases. 12. Cloning. .n ME=£xSEgn=ix/Ei=rawxf93§19%9z17%
n 1012 b) The pollsters are 95% confident that the true proportion of adults who approve of attempt
to clone a human is within 1.7% of the estimated 8%. c) A 90% confidence interval requires a smaller margin of error. If confidence is decreased, a
smaller interval is allowed. m ME=5>AEQD=EX/Ei=1eﬁx/gyﬁﬁggaz14%
n 1012 e) Smaller samples generally produce larger intervals. Smaller samples are more variable,
which increases the margin of error. 18. Drinking. a) [9 — g z 0.191. About 19.1% of this city’s youth reported having been drunk. 7 110 b) This estimate is from one sample. Other samples will give different proportions. We neei
to create a confidence interval. Plausible independence condition: There is no reason to believe that one randomly
selected student’s response will affect another’s. The survey was anonymous.
Randomization condition: The health agency randomly selected 110 respondents. 10% condition: 110 students is less than 10% of the 1212 students. ‘
Success/Failure condition: nfa= 21 and m}: 89 are both greater than 10, so the sample is large enough. C) Since the conditions are met, we can use a one—proportion zinterval to estimate the proportion of the city’s youth who have been drunk. (%>(%%)
110 13 i 3‘1 = i 1.960 n 110
We are 95% confident that between 11.7% and 26.4% of the city’s youth have been drunk. : (11.7%, 26.4%) There is reason to believe that the national level of 30% is not true of the middle school
students in this city. The national level of 30% is above the interval. 24. Back to campus. a) b) d) Plausible independence condition: Students’ decisions to return are independent.
Randomization condition: The students were randomly selected. 10% condition: 1644 students are less than 10% of all students. Success/Failure condition: 7116: 1644(074) = 1217 and nc}= 1644(026) = 427 are both greater
than 10, so the sample is large enough. Since the conditions are satisfied, we can use a one—proportion z—interval to estimate the
retention rate for college students nationwide. 13 42“ ﬂ = (0.74) i 2.326 W = (71.5%,76.5%)
n 1644 We are 98% confident that between 71.5% and 76.5% of all college students return to
college after their freshman year. If we were to select repeated samples like this, we’d expect about 98% of the confidence intervals we created to contain the true proportion of all college students return to college
after their freshman year. 26. Back to campus again. a) The confidence interval for the retention rate in private colleges will be narrower than t
confidence interval for the retention rate in public colleges, since it is based on a larger
sample. b) Plausible independence condition: The retention rates are likely to be independent.
Randomization condition: The students were randomly selected.
10% condition: 505 students are less than 10% of all students.
Success/Failure condition: n13= 505(O.719) = 363 and ncj= 505(0.281) = 142 are both grez than 10, so the sample is large enough. Since the conditions are satisfied, we can use a oneproportion z—interval to estimate th<
retention rate for public college students nationwide. 13 i z* «‘31 = (0.719) i 1.960 ———(0'719)(0'281) n = (68.0%,75.8%) We are 95% confident that between 68.0% and 75.8% of all public college students retui
college after their freshman year. c) A public college whose retention rate is 75% should not claim to do a better job of keep
freshman than other public colleges. Based on the confidence interval, the overall pub]
college retention rate could actually be higher than 75%, since the interval contains 750% 6. Teachers. a) Not correct. Actually, 9 out of 10 samples will produce intervals that will contain the mean
salary for Nevada teachers. Different samples are expected to produce different intervals. b) Correct! This is the one! C) Not correct. A confidence interval is about the mean salary of the population of Nevada
“teachers, not the salaries of individual teachers. 12. Parking. a) Randomization condition: The weekdays were not randomly selected. We will assume
that the weekdays in our sample are representative of all weekdays.
10% condition: 44 weekdays are less than 10% of all weekdays.
Nearly Normal condition: We don't have the actual data, but since the sample of 44
weekdays is fairly large it is okay to proceed. The weekdays in the sample had a mean revenue of $126 and a standard deviation in
revenue of $15. The sampling distribution of the mean can be modeled by a Student’s t
model, with 44 ~ 1 = 43 degrees of freedom. We will use a onesample tinterval with 90‘} confidence for the mean daily income of the parking garage. (By hand, use IZO : 1.684) b) y i r;;1[%) = 126 i r;3(%)x(1222, 129.8) c) We are 90% confident that the interval $122.20 to $129.80 contains the true mean daily
income of the parking garage. (If you calculated the interval by hand, using t; 2: 1.684 from the table, your interval will be (122.19, 129.81), ever so slightly wider from the interv
calculated using technology. This is not a big deal.) d) 90% of all random samples of size 44 will produce intervals that contain the true mean
daily income of the parking garage. e) Since the interval is completely below the $130 predicted by the consultant, there is
evidence that the average daily parking revenue is lower than $130. 16. Speed of Light. a) y i z;:_1[—j:) = 756.22 iz:2(1:)/72'—:2) z (709.9, 802.5) 1)) We are 95% confident that the interval 299,7099 to 299,802.5 km / sec contains the speed of
light. C) We have assumed that the measurements are independent of each other and that the
distribution of the population of all possible measurements is Normal. The assumption of
independence seems reasonable, but it might be a good idea to look at a display of the
measurements made by Michelson to verify that the Nearly Normal Condition is satisfied. 17. Second dog. a) b) d) This larger sample should produce a more accurate estimate of the mean sodium come
the hot dogs. A larger random sample has a smaller standard error, which results in a
smaller margin of error. S 32 .
SE ‘ = —— = W x 4.1 m sodium.
0’) l 8) g
Randomization condition: We don’t know that the hot dogs were sampled at random, 1
it is reasonable to think that the hot dogs are representative of hot dogs of this type.
10% condition: 60 hot dogs are less than 10% of all hot dogs. Nearly Normal condition: We don’t have the actual data, but since the sample of 60 hot
dogs is large it is okay to proceed. The hot dogs in the sample had a mean sodium content of 318 mg, and a standard
deviation in sodium content of 32 mg. Since the conditions have been satisfied, construr
onesample t—interval, with 60 ~ 1 = 59 degrees of freedom, at 95% confidence. y i z;;,.[%) z 318 i x (309.7, 326.3) We are 95% confident that the interval 309.7 to 326.3 mg contains the true mean sodium
content of this type of hot dog. If a “reduced sodium” hot dog has to have at least 30% less sodium than a hot dog
containing an average of 465 mg of sodium, then it must have less than 070(465) = 325.5
mg of sodium. Since our 95% confidence interval extends above this value, there is not
enough evidence to suggest that this type of hot dog can be labeled as “reduced sodium 18. Better light. a) b) __ i _ 79.0 =
Saw—[MJ—[fm] 7.9 km/sec. The interval should be narrower. There are three reasons for this: the larger sample size
results in a smaller standard error (reducing the margin of error), the larger sample size
results in a greater number of degrees of freedom (decreasing the value of ti, reducing tl
margin of error), and the smaller standard deviation in measurements results in a smalle
standard error (reducing the margin of error). Additionally, the interval will have a
different center, since the sample mean is different. We must assume that the measurements are independent of one another. Since the 8am}
size is large, the Nearly Normal Condition is overridden, but it would still be nice to lool
at a graphical display of the measurements. A one—sample l—interval for the speed of ligl
can be constructed, with 100 — 1 = 99 degrees of freedom, at 95% confidence. y i 5.671) = 852.4 i z;9(%) : (836.72, 868.08) We are 95% confident that the interval 299,836.72 to 299,868.08 km/ sec contains the speed
of light. Since the interval for the new method does not contain the true speed of light as reported
by Stigler, 299,710.5 km / sec., there is no evidence to support the accuracy of Michelson’s
new methods. The interval for Michelson’s old method (from Exercise 14) does contain the true speed of
light as reported by Stigler. There is some evidence that Michelson's previous
measurement technique was a good one, if not very precise. ...
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 Winter '08
 Ioudina
 Normal Distribution, Standard Deviation, Michelson, nearly normal condition

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