Section 9 - Breazeal



Section 11.3 – Inference About Two Means: Independent Samples

Objectives

1. Test hypotheses regarding the difference of two independent means

2. Construct and interpret confidence intervals regarding the difference of two independent means

Objective 1 – Test hypotheses regarding the difference of two independent means

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Example

Given the sample data below (assume that the populations are normally distributed)

| |Population 1 |Population 2 |

|n |20 |20 |

|[pic] |111 |104 |

|s |8.6 |9.2 |

Test whether (1 ( (2 at the ( = 0.05 level of significance. Either Reject H0 or Do Not Reject H0

Example

Given the sample data below (assume that the populations are normally distributed)

| |Population 1 |Population 2 |

|n |40 |32 |

|[pic] |94.2 |115.2 |

|s |15.9 |23.0 |

Test whether (1 < (2 at the ( = 0.05 level of significance. Either Reject H0 or Do Not Reject H0

Example

A researcher wanted to know whether “state” quarters had a weight that is more than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the following data.

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Test the claim that “state” quarters have a mean weight that is more than “traditional” quarters at the α = 0.05 level of significance.

NOTE: A normal probability plot of “state” quarters indicates the population could be normal. A normal probability plot of “traditional” quarters indicates the population could be normal

Example

For a sample of 1,657 men (ages 65-74), the mean weight is 164 lbs and the standard deviation is 27 lbs. For a sample of 804 men (ages 25-34), the mean weight is 176 lbs and the standard deviation is 35 lbs. Test the claim that the older men come from a population with a mean weight that is less than the mean weight for men in the younger age bracket. Use α = 0.01.

Example

A study of zinc-deficient mothers was conducted to determine whether zinc supplementation during pregnancy results in babies with increased weights at birth. Is there sufficient evidence at α = 0.05 to support the claim that zinc supplementation does result in increased birth weight?

(weights measured in grams) Zinc Supplement Placebo

n1 = 294 n2 = 286

[pic] = 3214 [pic] = 3088

s1 = 669 s2 = 728

Objective 2 - Construct and interpret confidence intervals regarding the difference of two independent means

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Given the sample data below (assume that the populations are normally distributed)

| |Population 1 |Population 2 |

|n |20 |20 |

|[pic] |111 |104 |

|s |8.6 |9.2 |

Construct a 95% confidence interval about (1 - (2 .

Example

Given the sample data below (assume that the populations are normally distributed)

| |Population 1 |Population 2 |

|n |40 |32 |

|[pic] |94.2 |115.2 |

|s |15.9 |23.0 |

Construct a 95% confidence interval about (1 - (2 .

Example

Construct a 95% confidence interval about the difference between the population mean weight of a “state” quarter versus the population mean weight of a “traditional” quarter (continued from example in objective 1).

Example

In a Gallup poll conducted August 32-September 2, 2009, 513 national adults aged 18 years of age or older who consider themselves to be Republican were asked “Of every tax dollar that goes to the federal government in Washington, DC, how many cents of each dollar are would you say are wasted?” The mean was found to be 54 cents with a standard deviation of 2.9 cents. The same question was asked of 513 Democrats. The mean was 41 cents with a standard deviation of 2.6 cents. Construct a 95% confidence interval for the mean difference in government waste, (R - (D and interpret the interval.

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Calculator Instructions

1. If necessary, enter raw data into L1 and L2

2. Press STAT, TESTS, and select 4: 2-SampTInt

3. Inpt: Data or Stats?

a. If data are raw, highlight Data, set List1 to L1 and List2 to L2, with frequencies set to 1

b. If summary statistics are known, highlight Stats and enter summary statistics

4. Enter the C-Level. Set Pooled to NO

5. Highlight Calculate and press ENTER

Steps for Hypothesis Testing Regarding the Difference of Means

1. Determine the null and alternative hypothesis

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2. Select a level of siggnificance, (

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

Calculator Instructions

6. If necessary, enter raw data into L1 and L2

7. Press STAT, TESTS, and select 4: 2-SampTTest

8. Inpt: Data or Stats?

a. If data are raw, highlight Data, set List1 to L1 and List2 to L2, with frequencies set to 1

b. If summary statistics are known, highlight STATS and enter summary statistics

9. Highlight the appropriate relationship between (1 and (2 in the alternative hypothesis. Set Pooled to NO

10. Highlight Calculate or Draw and press ENTER

Requirements

1. The samples are obtained using simple random sampling or through a randomized experiment

2. For each sample, the sample size is no more than 5% of the population size

3. The populations from which the samples are drawn are normally distributed or the samples are large (n1 ( 30 and n2 (30)

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