Statistics and Probability



Statistics

UNIT 9 - (2

Notes 9.1

Test for a Single Variance

So far, two distributions, the z and the t distributions have been explained. Another frequently used distribution is the chi-square ([pic]) distribution. This distribution can be used to test hypotheses concerning variances. The chi-square distribution is a family of curves based on the number of degrees of freedom.

Properties of Chi-Square Distribution

1. Chi-square variable cannot be negative.

2. Distributions are positively skewed.

3. At about 100 degrees of freedom, the chi-square distribution becomes somewhat symmetrical.

4. The area under each chi-square distribution is equal to 1 or 100%.

Ex. 1 Find the critical [pic] value for 15 degrees of freedom when [pic] = 0.05 and the test is one-tailed right.

Ex. 2 Find the critical [pic] value for 10 degrees of freedom when [pic] = 0.05 and the test is one-tailed left.

Ex. 3 Find the critical [pic] value for 22 degrees of freedom when [pic] = 0.05 and the test is two-tailed.

After the degrees of freedom reach 30, the table only gives values for multiples of 10. When the exact degrees of freedom are not specified in the table, the closest smaller value should be used.

Test Statistic

Variance or St. Deviation

[pic]

The hypothesis testing procedure remains the same!

Ex. 4 An instructor wishes to see whether the variation in IQ of the 23 students in her class is less than the variance of the population. The variance of the class is 198. Test the claim that the variation of the students is less than the population variance ([pic] = 225) at the 0.05 level of significance. Assume that IQ is normally distributed.

Ex. 5 A medical researcher believes that the standard deviation of the temperatures of newborn infants is greater than 0.6 degrees. A sample of 15 infants was found to have a standard deviation of 0.8 degrees. At the 0.10 level of significance, does the evidence support the researcher’s belief? Assume that the variable is normally distributed.

Ex. 6 A cigarette manufacturer wishes to test the claim that the variance of the nicotine content of the cigarette his company manufactures is equal to 0.644. Nicotine content is measured in milligrams; assume that it is normally distributed. A sample of 20 cigarettes has a standard deviation of 1.00 milligram. At the 0.05 level of significance, test the manufacturer’s claim.

Statistics

Notes 9.2

Confidence Intervals Variances

The chi-square distribution can also be used to find a specific confidence interval for the variance or standard deviation of a variable.

Confidence Interval Confidence Interval

Variance Standard Deviation

[pic] [pic]

Ex. 1 A cigarette manufacturer wishes to test the claim that the variance of the nicotine content of the cigarette his company manufactures is equal to 0.644. Nicotine content is measured in milligrams; assume that it is normally distributed. A sample of 20 cigarettes has a standard deviation of 1.00 milligrams. Find the 95% confidence interval of the variance and standard deviation for the nicotine content.

Ex. 2 Construct a 95% confidence interval about [pic] if a random sample of 21 scores is selected from a normally distributed population and the sample variance is 100.

Ex. 3 When working high school students are randomly selected and surveyed about the time they work after-school jobs, the mean and standard deviation are found to be 17.6 hours and 9.3 hours, respectively. Assume that this data is from a sample of 50 students and construct the 99% confidence interval about the standard deviation for all working high school students.

Statistics

Notes 9.3

Test for Goodness of Fit

The Chi-Square statistic can also be used to see whether a frequency distribution fits a specific pattern. In this case, the Chi-Square Goodness-of-Fit Test is used.

Formula for the Chi-Square Goodness-of-Fit Test

[pic] [pic]

Ex. 1 Suppose a market analyst wishes to see whether consumers have any preference among five flavors of a new fruit soda. A sample of 100 people provided the following data:

Cherry Strawberry Orange Lime Grape

32 28 16 14 10

Make a table with observed frequencies and expected frequencies:

|Frequency |Cherry |Strawberry |Orange |Lime |Grape |

| | | | | | |

| | | | | | |

Step 1: State the Hypotheses

Step 2: Find the critical value (Draw a curve)

Step 3: Compute the test statistic.

Step 4: Make the decision, and state the conclusion.

Ex. 2 The advisor of an ecology club at a large college believes that the group consists of 10% freshmen, 20% sophomores, 40% juniors, and 30% seniors. The membership for the club this year consisted of 11 freshmen, 19 sophomores, 47 juniors, and 23 seniors. Test the advisor’s conjecture at the 0.10 level of significance.

Statistics

Notes 9.4

Test for Independence

When data are tabulated in table form in terms of frequencies, a chi-square independence test can be used to test the independence of two variables. For example, suppose a new postoperative procedure is administered to a number of patients in a large hospital. One can ask whether doctors feel differently about this procedure than nurses do, or do they feel basically the same way? (Note that the question is not whether or not they prefer the procedure, but whether there is a difference of opinion between the two groups).

The data will be arranged in a contingency table. The table is made up of R rows and C columns.

|Group |Prefer New Procedure |Prefer Old Procedure |No Preference |

|Nurses |100 |80 |20 |

|Doctors |50 |120 |30 |

Find the value of the test statistic.

[pic] [pic]

Ex. 2 A sociologist wishes to see whether the number of years of college a person has completed is related to his or her place of residence. A sample of 70 is selected and classified as shown.

|Location |No College |B.S. Degree |Advanced Degree |Total |

|Urban |15 |12 |8 | |

|Suburban |8 |15 |9 | |

|Rural |6 |8 |7 | |

|Total | | | | |

At the 0.05 level of significance, can the sociologist conclude that the years of college education are dependent upon the residence location?

Ex. 3 A researcher wishes to determine whether there is a relationship between the gender of an individual and the amount of alcohol consumed. A sample of 68 people was selected, and the following data were obtained.

Alcohol Consumption

|Gender |Low |Moderate |High |Total |

|Male |10 |9 |8 | |

|Female |13 |16 |12 | |

|Total | | | | |

At the 0.10 level of significance, can the researcher conclude that alcohol consumption is related to the gender of the individual?

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