1. It is believed that \(40 \%\) of people pass their driving test on the first attempt. Suppose you think the percentage is greater than \(40 \%\). So, you perform a hypothesis test and sample \(100\) people. Of the sampled people, \(47\) reply that they passed on their first attempt. Set up a hypothesis test and make a conclusion with a \(10 \%\) significance level.

\(H_0: p \leq .40\)

\(H_A: p > .40\)

# 1. Draw a picture that includes the p-value, the observation, a title that describes the sampling distribution, and label the center

#2. Find the z-score or t-score of our observation

z <- (.47 - .40)/(sqrt(.4*.6/100))

#3. (optional) you can now draw a standardized version of your picture

#4. Find the p-value
1-pnorm(z)
## [1] 0.07652094
# Without R, find your z-score on the z-table (maybe you need to do 1-value)
1-.9236
## [1] 0.0764
# Make a conclusion
# Since our p-value of .0765 is smaller than the significance level of .1, we reject the null hypothesis
# Our data provides statistically significant evidence that more than 40% of people pass their driving test on the first attempt
  1. If the true proportion of people pass their driving test on the first attempt is \(52 \%\), what is the power of your test at detecting this difference?
qnorm(.9)
## [1] 1.281552
.4 + 1.28*sqrt(.4*.6/100)
## [1] 0.4627069
# Our cut-off value is .4627

# Find the z-score for your cut-off using the new curve
(.4627-.52)/(sqrt(.52*.48/100))
## [1] -1.146918
# The power is 88%, the probability of a type II error is 12%

  1. Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at \(0.5\) part per million (ppm). A hypothesis test is to be performed to decide whether the mean cadmium level in mushrooms is greater than the government’s recommended limit. It is known that the population is normally distributed and has standard deviation \(0.37\) ppm of cadmium. A sample of \(12\) mushrooms has a mean cadmium level of \(0.55\) ppm a significance level of \(5 \%\). Does the data provide statistically significant evidence that the mean cadmium levels is higher than the government’s recommend limit?

  2. If the true average is \(0.8\) ppm, what is the power of the test at detecting this?