Question

The power function for the one-sample \(t\) -test is discussed. (a) Plot the power function for the following setup: \(X\) has a \(N\left(\mu, \sigma^{2}\right)\) distribution; \(H_{0}: \mu=50\) versus \(H_{1}: \mu \neq 50 ; \alpha=0.05 ; n=25 ;\) and \(\sigma=10\). (b) Overlay the power curve in (a) with that for \(\alpha=0.01\). Comment. (c) Overlay the power curve in (a) with that for \(n=35\). Comment. (d) Determine the smallest value of \(n\) so the power exceeds \(0.80\) to detect \(\mu=53\). Hint: Modify the R function tpowerg.R so it returns the power for a specified alternative.

Short answer

Part (a): Power function plotted with given parameters. Part (b): Power increases with an increase in significance level. The \(\alpha = 0.01\) curve lies above the \(\alpha = 0.05\) curve. Part (c): Power increases with an increase in sample size. The \(n = 35\) curve lies above the \(n = 25\) curve. Part (d): The smallest \(n\) for power > 0.80 for \(\mu = 53\) is obtained by iterating over possible values of \(n\) until the power is greater than 0.80.

Step-by-step solution

Plot power function

Use the given parameters to create a function for the power of the test in R. The function can be plotted using the plot function in R - plot(x, power_fun(x), type='l'), where x is a range of values around the null hypothesis mean. The y-coordinate (power) for each x (alternative mean) can be obtained by using the function pt(), which gives the probability that a t-distributed random variable is less than x.

Overlay the curve for different significance level

Change the significance level (\(\alpha\)) to 0.01 in the function, and plot the new power function on the same plot. Compare the two curves to comment on the effect of the significance level on the power of the test. Generally, increasing the significance level would increase the power of the test.

Overlay the curve for different sample sizes

Next, change the sample size (\(n\)) to 35 in the function, and plot the new power function on the same plot. Compare the three curves to comment on how the sample size affects the power of the test. Generally, increasing the sample size will increase the power of the test.

Determine the smallest sample size for required power

To find the smallest sample size (\(n\)) that gives a power of more than 0.80 to detect a mean of 53, use a while loop in R to increment \(n\) until the function returns a power value of more than 0.80. Note that the 'tpowerg.R' function provided in the exercise might need to be manipulated to be used for this purpose.