# Statistical Sciences 1024A/B Chapter Notes - Chapter 16: Null Hypothesis, Standard Deviation, Binomial Distribution

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24 Sep 2015

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Department

Professor

172

Chapter 16 Solutions

16.1: The most important reason is (c); this is a convenience sample consisting of the first 20

students on a list. This is not an SRS. Anything we learn from this sample will not extend to the

larger population. The other two reasons are valid, but less important issues. Reason (a) — the

size of the sample and large margin of error – would make the interval less informative, even if

the sample were representative of the population. Reason (b) – nonresponse — is a potential

problem with every survey, but there is no particular reason to believe it is more likely in this

situation.

16.2: (a) The 95% confidence interval is

x

±

z*

σ

n

= 1.92

±

1.96

1.83

880

= 1.92

±

0.1209 =

1.80 to 2.04 motorists. (b) The large sample size means that, because of the central limit

theorem, the sampling distribution of

x

is roughly Normal even if the distribution of responses is

not. (c) Only people with listed telephone numbers were represented in the sample, and the low

response rate (10.9% =

5,029 45,956

) means that even that group may not be well represented

by this sample.

16.3: Any number of things could go wrong with this convenience sample. The day after

Thanksgiving is widely regarded (rightly or wrongly) as a day on which retailers offer great

deals — and the kinds of shoppers found that day probably don’t represent shoppers generally.

Also, the sample isn’t random.

16.4: (a) The 95% confidence interval is 159

±

1.96

35

501

= 159

±

3.06 = 155.94 pounds to

162.06 pounds. (b) The sample is an SRS, and it is certainly large enough so that the central

limit theorem applies — we know that

x

has a sampling distribution that is approximately

Normal. However, people are often inclined to misrepresent (lie or underestimate) their weights.

Gallup did not weigh these women — they reported their own weights. If their responses are

honest, then the confidence interval above can be relied upon. However, this may well not be the

case.

16.5: No. The confidence interval does not describe the range of future values of

x

. Instead, if

we repeated the experiment over and over, each time computing a 95% confidence interval for

the population mean

µ

, then 95% of these confidence intervals would capture the true population

mean.

16.6: With z* = 1.96 and

σ

= 75, the margin of error is z*

σ

n

=

14.7

n

. (a) If n = 100, the margin

of error is 1.47. (b) If n = 400, the margin of error is 0.735. If n = 1600, the margin of error is

0.3675. (c) As the sample size increases, the margin of error decreases. Notice that when the

sample size quadruples, the margin of error is halved.

16.7: The margin of error only addresses chance variation in the random selection of a sample.

Hence, the answer is (c). Sources of bias described in (a) and (b) are not accounted for in the

margin of error, and are difficult to assess.

Solutions 173

16.8: Computing by hand, we get results almost identical to those obtained using the Applet.

The standard deviation of

x

is

σ

n

=

116

50

= 16.405. The test statistic is

x −515

16.405

. (a) If

x

=

541, the test statistic is z = 1.58. The P-value is P =

P Z ≥1.58

( )

= 0.057. Hence, this is (barely)

not significant at the 5% level. (b) If

x

= 542, the test statistic is z = 1.65. The P-value is P =

P Z ≥1.65

( )

= 0.049. Hence, this is (barely) significant at the 5% level. Notice that there’s no

practical difference between

x

= 541 and

x

= 542, yet the decision we would make changes,

strictly applying the 5% significance level.

16.9: The full applet output for n = 5 is below on the left. Next to this are the Normal curves

drawn for n = 15 and n = 40. The reported P-values agree with the “hand-computed” values z =

4.8 −5

0.5 n

, and P =

P Z ≤z

( )

, given in the table.

16.10: The confidence intervals are given in the table to the right.

In each case, the interval is 4.8

±

1.96

0.5

n

.