Page 364 - DMTH404_STATISTICS

Statistics

Notes 2. It will be discussed in the following chapter that when expected value of a statistic equals
the value of parameter, it is said to be an unbiased estimate of the parameter.
Problem 1
The Acme Battery Company has developed a new cell phone battery. On average, the battery
lasts 60 minutes on a single charge. The standard deviation is 4 minutes.
Suppose the manufacturing department runs a quality control test. They randomly select 7
batteries. The standard deviation of the selected batteries is 6 minutes. What would be the
chi-square statistic represented by this test?

Solution
We know the following:
 The standard deviation of the population is 4 minutes.
 The standard deviation of the sample is 6 minutes.

 The number of sample observations is 7.
To compute the chi-square statistic, we plug these data in the chi-square equation, as shown
below.
2
2
x = [ ( n – 1 ) * s ] / s 2
2
2
x = [ ( 7 – 1 ) * 6 ] / 4 = 13.5
2
2
where x is the chi-square statistic, n is the sample size, s is the standard deviation of the sample,
and s is the standard deviation of the population.
Problem 2
Let’s revisit the problem presented above. The manufacturing department ran a quality control
test, using 7 randomly selected batteries. In their test, the standard deviation was 6 minutes,
which equated to a chi-square statistic of 13.5.
Suppose they repeated the test with a new random sample of 7 batteries. What is the probability
that the standard deviation in the new test would be greater than 6 minutes?
Solution
We know the following:

 The sample size n is equal to 7.
 The degrees of freedom are equal to n - 1 = 7 - 1 = 6.
 The chi-square statistic is equal to 13.5 (see Example 1 above).
Given the degrees of freedom, we can determine the cumulative probability that the chi-square
statistic will fall between 0 and any positive value. To find the cumulative probability that a
chi-square statistic falls between 0 and 13.5, insert the values in formula then result is the
cumulative probability: 0.96.
This tells us that the probability that a standard deviation would be less than or equal to 6
minutes is 0.96. This means (by the subtraction rule) that the probability that the standard
deviation would be greater than 6 minutes is 1 – 0.96 or .04.

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