You can test whether distribution parameters are equal to a specified value, such as parameters from an historical distribution.

A chi-square test will determine whether the distribution parameter is significantly different from the specified value. Compare the p-value with your pre-determined α-value.

- If the p-value is less than the α-value, then you can conclude that the distribution parameter is significantly different from the specified value.
- If the p-value is greater than the α-value, then you cannot conclude that the distribution parameter is significantly different from the specified value.

Test for Shape Equal to 5
Chi-Square DF P
22.3245 1 0.000

For the muffler data, the test is whether the shape parameter of the distribution for miles driven with the new type of mufflers is significantly different from `5`. According to historical data, the shape parameter is usually 5.

Because the p-value of `0.000` is less than an α-value of 0.05, you can conclude that the shape parameter is significantly different from 5. Thus, the shape parameter for the distribution of the new type of mufflers differs from the shape value in previous data.

Minitab also provides a Bonferroni confidence interval, which is associated with a chi-square test, to provide an interval of reasonable values for the parameter.

- When the chi-square test is significant (when you reject the null hypothesis that the parameter equals a specified value), the corresponding confidence interval will not contain the specified value.
- When the chi-square test is not significant (when you fail to reject the null hypothesis), then typically the corresponding confidence interval will contain the value specified for the test.

Bonferroni 95.0% (indiv 95.00%) Simultaneous CI
Shape parameter for
Variable Lower Estimate Upper
StartNew 5.436 5.768 6.120

For the muffler data, reasonable values for the shape parameter for the new type of mufflers are between 5.436 and 6.120. Notice that the interval does not contain 5 because the null hypothesis for shape = 5 was rejected.