Section 9 - Breazeal



Section 11.2 – Inference about Two Means: Dependent Matched Pairs

Objectives

1. Test hypotheses regarding matched-pairs data

2. Construct and interpret confidence intervals about the population mean difference of matched-pairs data

Objective 1 – Test hypothesis regarding matched-pairs data

Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean, except that the differences are analyzed.

Notation:

d = individual difference between two values in a matched pair

[pic]= mean value of the differences d for the paired sample data (average of “x – y” values)

sd = standard deviation of the differences d for the paired sample data

n = number of pairs of data

μd = mean value of the differences d for the population of paired data

The following requirements must be met before performing the hypothesis test:

[pic]

[pic]

[pic]

Example

A statistics student heard that an individual’s arm span is equal to the individual’s height. To test this hypothesis, the student used a random sample of 10 students and obtained the following data.

|Student |Height |Arm span |

|1 |59.5 |62 |

|2 |69 |65.5 |

|3 |77 |76 |

|4 |59.5 |63 |

|5 |74.5 |74 |

|6 |63 |66 |

|7 |61.5 |61 |

|8 |67.5 |69 |

|9 |73 |70 |

|10 |69 |71 |

Is the sampling method dependent or independent? Does the sample evidence contradict the belief that an individual’s height and arm span are the same at the ( = 0.05 level of significance?

Example

It is a commonly held belief that SUVs are safer than cars. If an SUV and car are in a collision, does the SUV sustain less damage as suggested by the cost of repair? The insurance Institute for Highway Safety crashed SUVs into cars. The SUV was moving 10 mph and the front of the SUV crashed into the rear of the car.

[pic]

Is this matched pairs data? Do the data suggest the repair cost for the car is higher at an ( = 0.05 level of significance?

Example

The following data represent the cost of a one-night stay in Hampton Inn Hotels and La Quinta Inn Hotels for a random sample of 10 cities. Test the claim that Hampton Inn Hotels are priced differently than La Quinta Hotels at the α = 0.05 level of significance.

|City |Hampton Inn |La Quinta |

|Dallas |129 |105 |

|Tampa Bay |149 |96 |

|St. Louis |149 |49 |

|Seattle |189 |149 |

|San Diego |109 |119 |

|Chicago |160 |89 |

|New Orleans |149 |72 |

|Phoenix |129 |59 |

|Atlanta |129 |90 |

|Orlando |119 |69 |

Objective 2 – Construct and interpret confidence intervals about the population mean difference of matched-pairs data

[pic]

The calculator will do all of this work for us.

[pic]

Example

Construct a 90% confidence interval for the mean difference in price of Hampton Inn versus La Quinta hotel rooms.

We are 90% confident that the mean difference in hotel room price for Ramada Inn versus La Quinta Inn is between $______ and $______.

Example

Reaction times for 6 drivers are recorded. Which of the following is a 90% confidence interval of the mean difference, μd? (di = xi – yi)

|With cell phone, xi |6.0 |4.5 |5.2 |3.9 |6.2 |4.1 |

|Without cell phone, yi |5.3 |4.0 |5.2 |4.1 |5.3 |3.8 |

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Point estimate ( margin of error

The point estimate is [pic].

The margin of error is given by

[pic], where the critical value [pic]is determined using n-1 degrees of freedom.

So the confidence interval looks like

[pic]

Calculator Instructions

1. If necessary, enter raw data in L1 and L2. Let L3 = L2 – L1, or L3 = L1 – L2, depending on how the alternative hypothesis is defined.

2. Press STAT, highlight TESTS, and select 8:TInterval

3. If the data are raw, select DATA, making sure that List is set to L3 with frequency set to 1. If summary statistics are known, highlight STATS and enter the summary statistics

4. Enter a confidence level, C-Level:

5. Highlight Calculate and press ENTER

To set L3 = L2 – L1, cursor to L3 column heading and enter the formula [2nd 2]− [2nd 1] ENTER

Steps for Hypothesis Testing Regarding Matched Pairs Data

1. Determine the null and alternative hypothesis

[pic]

2. Select a level of significance, (

3. Calculate the test statistic and P-value. These will come from the calculator

4. Compare P-value to (

If P–value < α, Reject H0

If P–value > α, Do Not Reject H0

5. State the conclusion

Requirements

• Samples are simple random samples

• Sample data consist of dependent matched pairs

• The number of matched pairs is greater than 30 or differences are normally distributed with no outliers.

• The values are independent, i.e., sample size no more than 5% of population size

Calculator Instructions

1. If necessary, enter raw data in L1 and L2. Let L3 = L2 – L1, or L3 = L1 – L2, depending on how the alternative hypothesis is defined.

2. Press STAT, highlight TESTS, and select 2:T-Test

3. If the data are raw, select DATA, making sure that List is set to L3 with frequency set to 1. If summary statistics are known, highlight STATS and enter the summary statistics

4. (0 will be 0

5. Highlight the appropriate relation in the alternative hypothesis

6. Highlight Calculate or Draw and press ENTER

To set L3 = L2 – L1, cursor to L3 column heading and enter the formula [2nd 2]− [2nd 1] ENTER

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