 # Putting Statistics to Work

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Putting Statistics to Work

Statistical Inference

Statistical Significance
A set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance. Common levels of significance: At the 0.05 level – The probability of an observed difference occurring by chance is 1 in 20 or less. At the 0.01 level – The probability of an observed difference occurring by chance is 1 in 100 or less. Copyright © 2011 Pearson Education, Inc.

Margin of Error and Confidence Intervals
Suppose you draw a single sample of size n from a large population and measure its sample proportion. The margin of error for 95% confidence is The 95% confidence interval is found by subtracting and adding the margin of error from the sample proportion. You can be 95% confident that the true population proportion lies within this interval. Discuss the relationship between the margin of error and the sample size. Copyright © 2011 Pearson Education, Inc.

Margin of Error and Confidence Intervals
Example: A survey of 1200 people finds that 47% plan to vote for Smith for governor. Find the margin of error. Find the 95% confidence interval for the survey. 47% – 2.9% = 44.1% % + 2.9% = 49.9% We can be 95% confident that the true proportion of people who plan to vote for Smith is between 44.1% and 49.9%. Discuss the relationship between the margin of error and the sample size. Copyright © 2011 Pearson Education, Inc.

null hypothesis: population parameter = claimed value
Hypothesis Testing The null hypothesis claims a specific value for a population parameter. It takes the form null hypothesis: population parameter = claimed value The alternative hypothesis is the claim that is accepted if the null hypothesis is rejected. Copyright © 2011 Pearson Education, Inc.

Outcomes of a Hypothesis Test
A hypothesis is a statement regarding a characteristic of one or more populations. Rejecting the null hypothesis → We have evidence that supports the alternative hypothesis. Not rejecting the null hypothesis → We lack sufficient evidence to support the alternative hypothesis. Copyright © 2011 Pearson Education, Inc.

Examples of Claims Regarding a Characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. Source: ReadersDigest.com poll created on 2008/05/02

Examples of Claims Regarding a Characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

Examples of Claims Regarding a Characteristic of a Single Population
In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

CAUTION! We test these types of statements using sample data because it is usually impossible or impractical to gain access to the entire population. If population data are available, there is no need for inferential statistics.

Alternative Hypothesis
The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support. In this chapter, it will be a statement regarding the value of a population parameter.

Forming Hypotheses For each of the following claims, determine the null and alternative hypotheses. In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

Solution a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. The hypothesis deals with a population proportion, p. If the percentage participating in charity work is no different than in 2008, it will be 0.62 so the null hypothesis is H0: p=0.62. Since the researcher believes that the percentage is different today, the alternative hypothesis is: H1: p≠0.62.

Solution b) b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. The hypothesis deals with a population mean. If the mean call length on a cellular phone is no different than in 2006, it will be 3.25 minutes so the null hypothesis is H0 =3.25. Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H1 > 3.25.

Solution c) c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased. The hypothesis deals with a population standard deviation. If the standard deviation with the new equipment has not changed, it will be 0.23 ounces so the null hypothesis is H0 = 0.23. Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H1 < 0.23.

CAUTION! We never “accept” the null hypothesis because without having access to the entire population, we don’t know the exact value of the parameter stated in the null hypothesis. Rather, we say that we do not reject the null hypothesis. This is just like the court system. We never declare a defendant “innocent”, but rather say the defendant is “not guilty”.

Stating Conclusions According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion. Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.

Solution a) Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion. The statement in the alternative hypothesis is that the mean call length is greater than 3.25 minutes. Since the null hypothesis (H0 = 3.25) is rejected, we conclude that there is sufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes.

Solution b) b) Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion. Since the null hypothesis (H0 = 3.25) is not rejected, we conclude that there is insufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes. In other words, the sample evidence is consistent with the mean call length equaling 3.25 minutes.

Hypothesis Test Decisions
Compare the actual sample result to the result expected if the null hypothesis is true. If the chance of a sample result at least as extreme as the observed result is less than 1 in 100 → strong evidence to reject the null hypothesis less than 1 in 20 → moderate evidence to reject the null hypothesis greater than 1 in 20 → not sufficient evidence to reject the null hypothesis Copyright © 2011 Pearson Education, Inc.

Hypothesis Test Example
Suppose the sample of 36 calls resulted in a sample mean of 3.56 minutes with a standard deviation of Do the results of this sample suggest that the researcher is correct? In other words, would it be unusual to obtain a sample mean of 3.56 minutes from a population whose mean is 3.25 minutes? What is convincing or statistically significant evidence?

Statistical Significance
When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

Hypothesis Test Example (cont.)
Recall that our simple random sample of 36 calls resulted in a sample mean of 3.56 minutes with standard deviation of Thus, the sample mean is standard deviations above the hypothesized mean of 3.25 minutes. Therefore, using our criterion, we would reject the null hypothesis and conclude that the mean cellular call length is greater than 3.25 minutes.

Standard Scores and Percentiles
Work through some simple concrete examples to demonstrate how the table works. Try different scenarios that require both adding and subtracting percentages. It may be helpful to emphasize the importance of sketching out a normal curve in the problem-solving process in order to visual what it is we are actually doing. Copyright © 2011 Pearson Education, Inc.

Hypothesis Test Example (cont.)
Why does it make sense to reject the null hypothesis if the sample mean is more than 2 standard deviations above the hypothesized mean?

Hypothesis Test Example (cont.)
If the null hypothesis were true, then 97.72% of all sample means will be less than 2 standard deviations above 3.25 or (0.13) = 3.51.

Hypothesis Test Example (cont.)
Because sample means greater than 3.51 are unusual if the population mean is 3.25, we are inclined to believe the population mean is greater than 3.25.

Assignment p. 410 – , 22, 27, 30, 33, 36, 39, 40, 45 , 46 Copyright © 2011 Pearson Education, Inc.