Significance testing: one-sided versus two-sided (3)

In the last exercises we saw that there are different cut-off values for one-sided and two-tailed hypothesis tests. You saw that in order to reject the null hypothesis when performing a two-tailed hypothesis, you would need to pass a higher threshold. In this exercise and the exercise to come, we will calculate probabilities based on sample means.

Let's go back to our example of scandinavian hipsters. Here we had a population mean of 25 and a population standard deviation of 3.5. Because we were taking samples of 40 subjects from this population, we were actually working with the standard error which was 3.5 / \(\sqrt{40}\). Imagine we found a sample mean of exactly 26 and a corresponding Z score of 1.81. Whether this result is significant depends on the test we do. If we would do a one-sided hypothesis test against a 5% significance level, we would only have to test for the effect in one direction. As such, we could check the following: \(P(>1.81)\). In order to do this, we could use our pnorm() function which calculates a probability that corresponds to a quantile or z score. We could use it in the following way: pnorm(1.81, lower.tail = FALSE). We set the lower tail equal to FALSE because pnorm() calculates the cumulative probability until the value of 1.81 and we want to know the probability of finding a value of 1.81 or larger. The functions yields a p value of 0.035 which is smaller than 0.05.

Imagine that we found a sample mean of 25.95 with a sample size of 40. Calculate the corresponding test statistic, a z score in this case, and assign it to the variable z_value. Look at the hint if you have forgotten the formula to calculate a z score. Assume that the population mean and standard deviation are the same as described above. Round all values in this exercise to two digits.
Use the pnorm() function to find the probability of finding a sample mean as large or more extreme and store this in the variable p_value. Round all values in this exercise to two digits.
Print the variable p_value to the console.

Take Hint (-30 XP)

Exercise

Significance testing: one-sided versus two-sided (3)

Instructions