10  Activity 7

We will first work with hypothesis testing by manual calculation before switching to the built-in functions.

10.1 Testing means

You are researching the effects of exercise on heart rate. In the general population, the mean bpm is 80. Among your sample of 36 people you find a mean of 75 bpm with a standard deviation of 10.

Exercise 10.1  

  1. Identify \(\mu_0, \bar x, n\) and \(s\) from the given information.
  2. Identify \(H_0\) and \(H_1\) for the claim that people who exercise regularly have a lower heart rate than the general population.
  3. Compute \(t = \dfrac{\bar x - \mu}{s / \sqrt n}\).
  1. Compute the threshold at the \(\alpha = 0.01\) level using the formula q = qt(1 - α, df=n-1)
  1. Because are testing for a lower bpm are we comparing \(t\) to \(q\) or to \(-q\)?
  2. Using the threshold, what is our conclusion about \(H_0\)?
  3. Compute the p-value using the formula pt(t, df=n-1)
  1. Based on the p-value would we reach the same conclusion about \(H_0\) if instead \(\alpha = 0.001\)? (I.e. if we wanted to be \(99.9\%\) confident)

10.1.1 Computing thresholds

Using the formula qt(1 - α, n - 1) or with 1 - α/2 for a two-sided threshold, compute both the one-sided and two-sided thresholds for:

Exercise 10.2  

  1. \(\alpha = 0.05, n = 25\)
  2. \(\alpha = 0.01, n = 50\)

Now, using your thresholds, test the following hypotheses. Do we reject or fail to reject \(H_0\)?

Exercise 10.3  

  1. \(H_1 : \mu > \mu_0\) with \(\alpha = 0.01, n = 50\) and test statistic \(t = 2.5\)
  2. \(H_1 : \mu \neq \mu_0\) with \(\alpha = 0.01, n = 50\) and test statistic \(t = 2.5\)
  3. \(H_1 : \mu < \mu_0\) with \(\alpha = 0.05, n = 25\) and test statistic \(t = -1.6\)
  4. \(H_1 : \mu < \mu_0\) with \(\alpha = 0.05, n = 25\) and test statistic \(t = -1.9\)
Tip

Recall: values that do not exceed the threshold are “usual” and we should not reject \(H_0\). Values that exceed the threshold are “unusual” and we reject \(H_0\).

10.1.2 Computing p-values

  • For \(P(T \le t)\) we use pt(tscore, df)
  • For \(P(T \ge t)\) we use 1 - pt(tscore, df)
  • For a two-sided test, we double this value and the easiest way to compute it is to take the negative value of the tscore and use 2*pt(tscore, df)

For the following situations, compute the p-value.

Exercise 10.4  

  1. \(H_1 : \mu > \mu_0\) with \(n = 50\) and test statistic \(t = \dfrac{\bar x - \mu}{s / \sqrt n} = 2.5\)
  1. \(H_1 : \mu \neq \mu_0\) with \(n = 50\) and test statistic \(t = 2.5\)
  1. \(H_1 : \mu < \mu_0\) with \(n = 25\) and test statistic \(t = -1.6\)
  1. \(H_1 : \mu < \mu_0\) with \(n = 25\) and test statistic \(t = -1.9\)

10.2 Hypothesis testing for proportions

Approximately 10% of people are left-handed. This proportion is higher in certain populations. In a sample of \(100\) people with dyslexia, \(20\) are left-handed

Exercise 10.5  

  1. Identify \(p_0, \hat p, \hat q\) and \(n\) from the given information.
  2. Identify \(H_0\) and \(H_1\) for the claim that the proportion of left-handedness is greater among people with dyslexia than the general population.
  3. Compute \(z = \dfrac{\hat p - p_0}{\sqrt{\hat p \hat q / n}}\).
  1. Compute the threshold at the \(\alpha = 0.01\) level using the formula q = qnorm(1 - α)
  1. Because are testing for a higher proportion are we comparing \(z\) to \(q\) or to \(-q\)?
  2. Using the threshold, what is our conclusion about \(H_0\)?
  3. Compute the p-value using the formula \(P(Z \ge z) = 1 - \mathtt{pnorm}(z)\)
  1. Based on the p-value would we reach the same conclusion about \(H_0\) if instead \(\alpha = 0.001\)? (I.e. if we wanted to be \(99.9\%\) confident)

10.2.1 Computing thresholds

Using the formula qnorm(1 - α) or with 1 - α/2 for a two-sided threshold, compute both the one-sided and two-sided thresholds for:

Exercise 10.6  

  1. \(\alpha = 0.05\)
  2. \(\alpha = 0.01\)

Now, using your thresholds, test the following hypotheses. Do we reject or fail to reject \(H_0\)?

Exercise 10.7  

  1. \(H_1 : \mu > \mu_0\) with \(\alpha = 0.01\) and test statistic \(z = 2.5\)
  2. \(H_1 : \mu \neq \mu_0\) with \(\alpha = 0.01\) and test statistic \(z = 2.5\)
  3. \(H_1 : \mu < \mu_0\) with \(\alpha = 0.05\) and test statistic \(z = -1.6\)
  4. \(H_1 : \mu < \mu_0\) with \(\alpha = 0.05\) and test statistic \(z = -1.9\)

10.2.2 Computing p-values

Exercise 10.8  

  1. \(H_1 : \mu > \mu_0\) and test statistic \(z = 2.5\)
  1. \(H_1 : \mu \neq \mu_0\) and test statistic \(z = 2.5\)
  1. \(H_1 : \mu < \mu_0\) and test statistic \(z = -1.6\)
  1. \(H_1 : \mu < \mu_0\) and test statistic \(z = -1.9\)

10.3 Using software (two-sample hypotheses)

Suppose we have some data about rates of seasonal depression in New York versus in California. Suppose \(5\%\) of people (\(x = 62, n = 1240\)) report seasonal depression in New York versus \(3\%\) (\(x = 33, n = 1100\)) in California. We wish to test the claim that the more northern state (New York) has a greater incidence of seasonal depression. We do this with the following R function.

Exercise 10.9  

  1. Write the hypotheses \(H_0\) and \(H_1\) for our claim about \(p_1\) and \(p_2\).
  2. From the output, what is the p-value for the claim?
  3. At the \(\alpha = 0.01\), does the p-value suggest evidence for the claim that New Yorkers experience a greater incidence rate for seasonal depression? I.e. do we reject \(H_0\)?

A study on the heart rate of (cisgender) men and women collects some data on the resting heart rate of each group of people. The statistician overseeing the project runs the following code to test the hypothesis that these two groups have a different resting heart rate.

Exercise 10.10  

  1. Write the hypotheses \(H_0\) and \(H_1\) for our claim about \(\mu_1\) and \(\mu_2\).
  2. What is the mean heart rate for the men in the sample? For the women?
  3. From the output, what is the p-value for the claim?
  4. At the \(\alpha = 0.01\), does the p-value suggest evidence for the claim that the two groups have a different resting heart rate?