A team of scientists are conducting research on the effect of word fonts used in road signs on accident rates. Across the popula
tion, there is an average of μ = 40 accidents per year due to unclear road sign fonts, with a standard deviation of σ = 10. The distribution is approximately normal. The team believes that certain fonts are difficult to read and may cause accidents, and it wants to implement a change to road sign fonts nationwide. The team is recommending a new font to replace the old fonts in road signs. After a series of studies, the team finds that the new font decreases accidents. However, the effect size is rather small (d = - 0.2). If the change to road sign fonts is implemented and there truly is an effect, it is expected that the new average rate of accidents will be: μ = 38 (i.e., hypothesized mean of the treatment group). In other words, this change should decrease accidents by 2, on average. [50 points total]
A.) Because implementing such large-scale changes is very costly, the government wants to be sure that the investment will be impactful. For a hypothesis test, they want the α-level to be small to minimize error. It is decided that α = 0.01, and a one-tailed test will be used. [50 points]
-What type of error is minimized by setting a stricter (smaller) α-level?
-With a one-tailed hypothesis test at α = 0.01, use the unit normal table to find the zscore that marks the critical region.
B.) A sample of n = 100 people are selected to go through a road trial with new fonts in the road signs. Again, the treatment is expected to decrease accidents by 2 on average. If the researchers use a one-tailed hypothesis test with α = 0.01, what is the power of the hypothesis test? [10 points]
C.)An external consultant is worried that there may not be enough power to detect such a small effect with such a small sample size. The consultant recommends that the researchers select a sample of n = 400 people instead. With this new sample size, and assuming a one-tailed hypothesis test with α = 0.01 again, what is the power of the hypothesis test?
D.)Based on your answers in 3(b) and 3(c), calculate the probability of making a Type II error in both cases and explain what the values mean.
E.) Explain how sample size influences power. Discuss what this means in terms of the conclusions you might make in hypothesis testing. (You can use your answers in 3(b) and 3(c) as an example.)
