Practice Problems # 07 – Solutions
1. Five measurements are taken of the octane rating for a particular type of gasoline. Assume
these are a simple random sample from the population of this type of gasoline, and that the
distribution of measurements is a normally distributed population. The results (in%) are 87.0,
86.0, 86.5, 88.0, 85.3; find a 99% two-sided tolerance interval on 90% of the population.
From my handout which was attached to lecture 13 notes, k = 6.612. Hence the tolerance
limits are given by:
(
)
(
)()
86.56 6.612 1.021274 86.56 6.753 79.81,93.31xks±= ± = ± =
2. For the data in problem number 3, find a 98% confidence interval for the standard deviation of
the population of measurements.
1.021274s=, 2
0.99,4 0.297109
χ
=, and 2
0.01,4 13.276704
χ
=
() () ()( )()( )
()( )()( ) ()
22
22
22 2 2
2, 1 2, 0.01,4 0.99,4
22
1 1 5 1 1.021274 5 1 1.021274
,,
5 1 1.021274 5 1 1.021274
, 0.561,3.747
13.276704 0.297109
rr
ns ns
αα
χχ χ χ
−
⎡⎤
⎡⎤
−− − −
⎢⎥
⎢⎥
=⎢⎥
⎢⎥
⎣⎦
⎣⎦
⎡⎤
−−
⎢⎥
=
⎢⎥
⎣⎦
3. Compounds of mercury and mercury ions are discharged into the atmosphere when coal is
burned in large quantities, as in a power generation station. These are then transported to the
land surface and water ways by rain, and of course, eventually make their way into the water
supply. Let μ denote the true average mercury level in the James River as measured in ppm. A
value of 10 ppm is considered the dividing line between safe and unsafe water levels of
mercury. Would you recommend testing H0: μ = 10 versus Ha: μ > 10 or H0: μ = 10 versus Ha:
μ < 10? Explain your reasoning. Think about the consequences of the type I and II errors for
each possibility.
Section 8.1 -- more on how to set null and alternative hypotheses
Suppose we specify the null hypothesis to be H0: μ = 10 (or H0: μ > 10), and the alternative is Ha:
μ < 10. This formulation is such that the water is believed unsafe (μ > 10) until proved otherwise
(μ < 10). A type I error involves deciding that the water is safe (rejecting H0) when it isn’t (H0 is
true). This is a very serious error, so a test that ensures that this error is highly unlikely is
desirable (i.e. we can require a very small α-value). A type II error involves judging the water
unsafe when it actually is safe. Though a serious error, this is less so than the type I error.
Suppose we specify the null hypothesis to be H0: μ = 10 (or H0: μ < 10), and the alternative Ha:
μ > 10. The type II error (now stating that the water is safe when it isn’t) is the more serious of
the two errors.
