A significance test examines whether the null hypothesis provides a plausible explanation of the data. The **null hypothesis** itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The **alternative hypothesis** is typically the research hypothesis of interest. Here are some examples.

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Example 10.2: Hypotheses with One Sample of One Categorical Variable
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About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

**Research Question**:*Are artists more likely to be left-handed than people found in the general population?***Response Variable**: Classification of the student as either right-handed or left-handed

##### State Null and Alternative Hypotheses

**Null Hypothesis**: Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or*p*= .10).**Alternative Hypothesis**: Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or*p*> .10). This is a one-sided alternative hypothesis.

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Example 10.3: Hypotheses with One Sample of One Measurement Variable
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A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

**Research Question**:*Does the data suggest that the population mean dosage of this brand is different than 50 mg?***Response Variable**: dosage of the active ingredient found by a chemical assay.

##### State Null and Alternative Hypotheses

**Null Hypothesis**: On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).**Alternative Hypothesis**: On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

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Example 10.4: Hypotheses with Two Samples of One Categorical Variable
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Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

**Research Question**:*Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?***Response Variable**: Classification of whether or not a person eats vegetarian meals on a regular basis**Explanatory (Grouping) Variable:**Sex

##### State Null and Alternative Hypotheses

**Null Hypothesis**: There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or*p*_{females}=*p*_{males}).**Alternative Hypothesis**: Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or*p*_{females}>*p*_{males}). This is a one-sided alternative hypothesis.

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Example 10.5: Hypotheses with Two Samples of One Measurement Variable
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Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

**Research Question**:*Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?***Response Variable**: Weight loss (pounds)**Explanatory (Grouping) Variable**: Type of diet

##### State Null and Alternative Hypotheses

**Null Hypothesis**: There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).**Alternative Hypothesis**: The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

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Example 10.6: Hypotheses about the relationship between Two Categorical Variables
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**Research Question**:*Do the odds of having a stroke increase if you inhale**second hand smoke**? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.***Variables**: There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.

##### State Null and Alternative Hypotheses

**Null Hypothesis**: There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).**Alternative Hypothesis**: There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

**Note!**

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

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Example 10.7: Hypotheses about the relationship between Two Measurement Variables
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**Research Question**:*A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day*(stock trading by management being under "insider-trading" regulatory restrictions).**Variables**: Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.

##### State Null and Alternative Hypotheses

**Null Hypothesis**: The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.**Alternative Hypothesis**: The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

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Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples
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**Research Question**:*Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?***Variables**: There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.

**State Null and Alternative Hypotheses**

**Null Hypothesis**: The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.**Alternative Hypothesis**: The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.