Type I and Type II Errors
Define a Type I error in your own words.
Rejecting the null hypothesis when it is actually true (a false positive).
What does a p-value represent in a hypothesis test?
The probability of observing results as extreme or more extreme than what was observed, assuming the null hypothesis is true.
What does a confidence interval estimate?
It estimates a range of plausible values for a population parameter based on sample data.
The statement that we assume to be true in a hypothesis test is called the ____.
Null hypothesis
The distribution we use when conducting hypothesis tests for means with small samples is called the ____.
t-distribution
Define a Type II error in your own words.
Failing to reject the null hypothesis when it is actually false (a false negative)
If a p-value is less than alpha, what should you do?
Reject the null hypothesis; there is enough evidence to support the alternative hypothesis.
If a 95% confidence interval for a proportion is (0.48, 0.56), what does that mean in context?
We are 95% confident that the true population proportion lies between 0.48 and 0.56.
What is the purpose of a hypothesis test for a proportion?
To determine if there is statistical evidence that the population proportion differs from a claimed value.
In a hypothesis test for a mean, the symbol for the population mean is ____.
mu (or μ)
A smoke detector goes off when there is no fire. Is this a Type I or Type II error? And WHY?
This is a Type I error. In this context, the null hypothesis would be that there is no fire, and the alarm going off is like rejecting that null hypothesis. Since the null was actually true (no fire), but the alarm falsely indicated danger, it's a Type I error.
A study reports a p-value of 0.04 and the significance level is 0.01. What decision should be made and why?
Fail to reject the null hypothesis, because the p-value (0.04) is greater than alpha (0.01); not enough evidence to support the alternative hypothesis.
What happens to the width of a confidence interval if you increase the confidence level from 95% to 99%?
The interval gets wider to reflect more certainty.
In a hypothesis test for a proportion, what does the p-value tell you?
It tells you the probability of observing a sample proportion as extreme or more extreme than the one observed, assuming the null hypothesis is true.
Why do we use the t-distribution instead of the normal distribution when working with means?
Because we are estimating the population standard deviation from the sample, and the t-distribution accounts for additional variability that happens in a smaller sample size
A school principal is testing whether the majority of students are following the new dress code policy. The null hypothesis is that most students are following the dress code.
Describe what a Type I error and a Type II error would look like in this situation.
Type I error: Concluding students are not following the dress code when they actually are.
Type II error: Concluding students are following the dress code when they actually are not.
How is the confidence level of a confidence interval related to alpha in a hypothesis test?
The confidence level is equal to 1−α. For example, if alpha is 0.05, the confidence level is 95%.
Name two factors that affect the margin of error in a confidence interval.
The confidence level and the sample size.
If the test statistic for a one-proportion z-test is z = 2.1, and α=0.05, what decision should you make using the critical value method?
Since the critical value for a one-tailed test at α=0.05 is approximately 1.645, and 2.1>1.645 you reject the null hypothesis.
You have a sample of 20 students, with a mean score of 82 and a standard deviation of 5. Find the standard error of the mean.
1.118
A pharmaceutical company is testing a new drug. The null hypothesis is that the new drug has no effect.
Describe what a Type I error and a Type II error would look like in this scenario, and explain which error would likely be more serious and why.
Type I error: Concluding the drug works when it actually doesn’t (approving an ineffective drug).
Type II error: Concluding the drug doesn’t work when it actually does (missing out on a helpful treatment).
More serious: The Type I error is more serious here, because it could lead to releasing an ineffective or unsafe drug to the public.
A hypothesis test is being conducted with α=0.01. Find the critical z-values for a two-tailed test.
The critical z-values are −2.576 and 2.576 for a two-tailed test at α=0.01
A poll of 400 people found that 58% support a new law. Find the approximate margin of error for a 90% confidence interval. Round to 4 decimal places.
0.0406
A researcher tests if the proportion of college students who drink coffee is greater than 50%. A sample of 200 students finds 116 who do. State the null and alternative hypotheses, and explain whether you would use a one- or two-tailed test.
Null hypothesis: H0:p=0.5
Alternative hypothesis: Ha:p>0.5
This is a one-tailed test because the researcher is only looking to see if the proportion is greater than 50%.
A study tests whether the mean number of hours students sleep is less than 7 hours. The sample mean is 6.8, standard deviation 0.5, and sample size 25. Write the null and alternative hypotheses and state whether this is a one-tailed or two-tailed test.
Null hypothesis: H0:μ=7
Alternative hypothesis: Ha:μ<7
This is a one-tailed test (since it’s testing if the mean is less than 7 hours).