Statistics 20: Quiz 3 Solutions
Summer Session 2007
1. The nicotine content in cigarettes of a certain brand is normally distributed with standard
deviation σ = 0.1 milligrams. The brand advertises that the mean nicotine content of their
cigarettes is µ = 1.5. The FDA believes that the actual nicotine content is higher than what
is advertised by the brand, and they plan to investigate the advertised claim by measuring
the nicotine content of 15 randomly selected cigarettes of this brand.
(a) [2 point] Give the appropriate null hypothesis H0 and alternative hypothesis Ha for the
test that the FDA plans to conduct.
Solution: Here the FDA wants to test whether the actual nicotine content is higher
than what is advertised by the brand which is 1.5 milligrams. So the hypotheses for the
test are,
H0 : µ = 1.5
Ha : µ > 1.5
(b) [3 points] Let x̄ denote the average nicotine content for the 15 cigarettes that were randomly sampled. If x̄ = 1.55, what is the value of the test statistic for testing your
hypotheses in (a)? (show all work)
Solution: Here x̄ = 1.55, σ = 0.1 and n = 15. So the value of the test statistic z =
is
z=
x̄−1.5
√
σ/ n
0.05 × 3.873
1.55 − 1.5
√
=
= 1.9365.
0.1
0.1/ 15
(c) [3 points] What is the P −value for your test statistic? Can you reject the null hypothesis
at the 5% significance level? (show all work)
Solution: The P-value for the test statistic is P(Z > z) = P(Z > 1.9365) = 0.0264.
Since the P-value is less than 5% we reject the null hypothesis at 5% significance level.
(d) [3 points] What is the smallest value of x̄ for which you will reject H0 at the 5% significance level?
Solution: The smallest value of x̄ for which we will reject H0 at the 5% significance
1
level is the value for which the P-value is 0.05. i.e.
x̄ − 1.5
√
P Z>
= 0.05
σ/ n
x̄ − 1.5
√ = 1.645
⇒
0.1/ 15
0.1
⇒ x̄ = 1.5 + 1.645 × √ = 1.5425.
15
2. Suppose that the readings of a laboratory scale are normally distributed with unknown mean
µ and standard deviation σ = 0.01 grams. To assess the accuracy of the laboratory scale,
a standard weight that is known to weigh exactly 1 gram is repeatedly weighed a total of
n = 50 times. Let x̄ = 0.998 be the average of the 50 readings.
You can use the following formulas regarding confidence intervals to answer the questions
below:
For a random sample of sizen drawn from a population
of unknown mean µ and known
σ
σ
∗
∗
√
√
SD σ, the (1 − α) CI for µ is x̄ − z n , x̄ + z n , where z ∗ is the critical value, selected
so that a standard Normal density has area (1 − α) between −z ∗ and z ∗ .
(a) [3 points] What is a 95% confidence interval for µ? (show all work)
Solution: The 95% critical
value for normal distribution is z ∗= 1.96. So a 95% confi
0.01
0.01
dence interval for µ is 0.998 − 1.96 × √
= (0.99523, 1.00077).
, 0.998 + 1.96 × √
50
50
(b) [3 points] What is a 99% confidence interval for µ? (show all work)
Solution: The 99% critical
value for normal distribution is z ∗ =2.576. So a 99% confi
0.01
0.01
dence interval for µ is 0.998 − 2.576 × √
= (0.99436, 1.00164).
, 0.998 + 2.576 × √
50
50
(c) [3 points] How large of a sample is needed so that the margin of error for a 95% confidence interval for µ is no larger than 0.0001. (show all work)
Solution: With a sample of size n margin of error for a 95% confidence interval for
µ is 1.96 × √σn . So to have a margin of error no larger than 0.0001 we should have
1.96 × √σn ≤ 0.0001, i.e.
n≥
1.96 × σ
0.0001
2
=
1.96 × 0.01
0.0001
2
= 38416.
3. Answer each of the following questions with True, False, or Unable to Determine
2
(a) [1 point]The margin of error for a 95% confidence interval for the population mean µ
increases as the sample size increases.
Solution: FALSE.
[Recall, the margin of error is
z√∗ σ
n
where z ∗ is the 1 − α critical value.]
(b) [1 point] The margin of error for a confidence interval for the population mean µ, based
on a specified sample size n, increases as the confidence level decreases.
Solution: FALSE.
(c) [1 point] The margin of error for a 95% confidence interval for the population mean µ
decreases as the population standard deviation decreases.
Solution: TRUE.
(d) [1 point] The sample size required to obtain a confidence interval of a specified margin
of error m, increases as the confidence level increases.
Solution: TRUE.
(e) [1 point] Assume that it is of interest to test H0 : µ = µ0 vs. Ha : µ 6= µ0 . If µ0 does
not fall within the two-sided 95% confidence interval for µ, then H0 can not be rejected
at the .05 significance level.
Solution: FALSE.
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