Definition and Description of Statistics
Define / describe in your own words the following terms and give an example of each.
a. F distribution
b. F statistic
c. Chi-square distribution
d. T distribution
e. Dependent samples
f. Independent samples
g. Degrees of freedom
h. T statistic
i. Paired difference
...ndard deviation in each sample and in each population.
Population Population standard deviation Sample standard deviation
Women 30 35
Men 50 45
Compute the f statistic.
c. Chi square distribution:
We select a random sample of size n from a normal population, having a standard deviation equal to σ. We find that the standard deviation in our sample is equal to s. Given these data, we can compute a statistic, called chi-square, using the following equation:
Χ2 = [ ( n - 1 ) * s2 ] / σ2
If we repeated this experiment an infinite number of times, we could obtain a sampling distribution for the chi-square statistic. The chi-square distribution is defined by the following probability density function:
Y = Y0 * ( Χ2 ) ( v/2 - 1 ) * e-Χ2 / 2
where Y0 is a constant that depends on the number of degrees of freedom, Χ2 is the chi-square statistic, v = n- 1 is the number of degrees of freedom, and e is a constant equal to the base of the natural logarithm system (approximately 2.71828). Y0 is defined, so that the area under the chi-square curve is equal to one.
Example: 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?
d. t distribution:
But sample sizes are sometimes small, and often we do not know the standard deviation of the population. When either of these problems occur, statisticians rely on the distribution of the t statistic (also known as thet score), whose values are given by:
t = [ x - μ ] / [ s / sqrt( n ) ]
where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n ...