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Name: September 30, 2011

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Lab 5: Chapter 6

READ THIS: In the material for the first exam, especially Sections 3.3 and 5.2, you learned about the difference between a population and a sample. There is a parameter for the whole population (for example the average age for all U.S. residents), but it would be too difficult, time-consuming, and expensive for us to get the number directly through a census. Instead, we usually choose to take a random sample and calculate a statistic as an estimate for the population parameter. However, we also know that every time we take a random sample from the population, we will get a different sample statistic simply due to random variability. When we use a statistical technique called inference, we are taking into account random variability when we estimate the population parameter from the sample statistic. Confidence intervals and hypothesis testing are types of inference.

1. (3 points) Using the diagram below, fill in the blanks with these terms: inference, sampling, (, σ, s, [pic].

[pic]

Confidence Intervals

Using Internet Explorer, go to the website for your textbook: ips6e. Click on “statistical applets” and then on “confidence interval.” Read the explanation at the top of the webpage. Use the 95% confidence level until you are told to change.

2. (0.5 points) What does the ( stand for? ___________________________

3. (0.5 points) When we calculate a confidence interval in real-life (non-applet) circumstances, do we know what ( is? Circle your answer: Yes No

Press the “sample” button to create a single confidence interval. Did your confidence interval from your single sample include the true population mean?

4. (1 point) What does the red slash in the middle of the confidence interval represent? (Answer with a symbol and in words.)

Try creating some more confidence intervals by hitting the “sample” button several more times. Check to see if all of your confidence intervals include the true population mean. If you create 100 confidence intervals, what percentage of your confidence intervals do contain the true population mean?

Try changing the confidence level and observe the results to your intervals.

5. (0.5 points) As the confidence level increases, the width of the interval (circle one):

increases / decreases / stays the same

6. (0.5 points) As the confidence level increases, the percentage of intervals which contain the true population mean (circle one):

increases / decreases / stays the same

This applet doesn’t let us adjust the sample size for our individual simple random samples. The formula for the confidence interval for the population mean when the population standard deviation is known is:

[pic]

7. (0.5 points) As the sample size increases, the width of the interval (circle one):

increases / decreases / stays the same.

8. (0.5 points) As the sample size increases, the center of the interval generally (circle one):

moves right / moves left / stays the same.

Hypothesis testing

READ THIS: For hypothesis testing, we don’t know the true population mean, but we do have a reasonable guess, possibly from previous research, which we call (0. This number goes in the null hypothesis. The point of hypothesis testing is to see how the true population mean compares to this reasonable guess without having to take a full census. Since we don’t know what the population mean really is, we use a simple random sample and calculate a sample statistic. However, we know that every time we take a sample, there is always random variability, so we need to take that into account when we estimate our population parameter. This is why you don’t just compare the sample statistic directly to (0. Instead, we need to see how unusual our sample statistic is if the null hypothesis is true. If the sample statistic isn’t too many standard deviations away from (0, then random chance could explain any discrepancy, and we have no reason to reject the null hypothesis. If the sample statistic is very far away from (0, then we say that our results are just too weird to have occurred simply due to random variation alone, and there is something wrong with our null hypothesis. How weird is too weird? We use a cut-off point, called the significance level ((, usually 0.05). If the P-value is less than the significance level, meaning results as weird as yours have a very small probability of occurring due to random chance, you reject the null hypothesis.

9. (4 points) Fill in the chart below for how we make decisions about conclusions to hypothesis tests. Your answer choices are in boldface type.

| |P-value > ( |P-value ( ( |

|a) Is the sample mean [pic]near or far from the reasonable guess for the| | |

|population mean (0? | | |

|b) On the Normal curve diagram for this hypothesis test, does the | | |

|P-value take up a small or big area? | | |

|c) Conclusion to the hypothesis test: | | |

|Reject H0, | | |

|Do not reject H0, or | | |

|Accept H0 | | |

|d) There is or is not evidence in favor of the alternative hypothesis. | | |

The mean weight for a simple random sample of 12 Hostess Twinkie snack cake packs is found to be 15.96 oz. The overall population standard deviation of a pack of Hostess Twinkies is 0.15 oz. The package promises a weight of 16 oz. Is the true population mean weight of a Twinkie package different from 16 oz? Use a 5% significance level.

10. (6 points) For the story above, state the hypotheses, calculate the test statistic and P-value (using the Normal table from the course webpage or your book), and state your conclusion in terms of the story. Be sure to state whether your conclusion refers to the population or the sample.

11. (1.5 points) Neatly label the Normal curve below with the number and symbol for the mean you use in your null hypothesis ((0) and the sample mean ([pic]). Also shade and label the appropriate part of the curve which shows the P-value.

[pic]

12. (1.5 points) Neatly label the Normal curve below with the number and symbol for z = 0, and the test statistic (z0). Also shade and label the appropriate part of the curve which shows the P-value.

[pic]

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population

sample

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