Define "sampling error"
A difference between a sample statistic and the corresponding pop. parameter
Why do we care about sampling error in the context of hypothesis testing?
Hypothesis testing is trying to determine if an observed difference between a statistic and parameter is the result of a sampling error (error in our study)
What is probability sampling?
A method of sampling that allows us to specify for each case, the probability of it's inclusion in the sample
1. Ensures every member of a pop. has an equal chance of being chosen
2. Ensures every combination of N members has an equal chance of being chosen
What does the sampling distribution of the mean look like? How do we know that?
Normal in shape, symmetrical, and it's mean is equal to the mean of the pop. from which it was drawn.
We know this b/c the central limit theorem states that the sample means for any pop. will be distributed roughly as a normal distribution around the pop. mean
Briefly explain the main point of Chebyshev's Theorem
Every normal distribution has a constant relationship with it's standard deviation. The 68% - 95% - 99.7% rule.
What is a type I error?
The null hypothesis is true but we accidentally reject it
Why is a type I error a bigger concern than a type II error?
Stating that a result is statistically significant when it's not can significantly impact research and beliefs on the given topic and change scientific assumptions. The biggest concern with a type II error is that a result is significant, but overlooked. (think the autism and vaccine study from Andrew Wakefield)
In hypothesis testing, was does an alpha level of 0.05 mean?
The max. probability of committing a type I error is 0.05, or, the probability of seeing a difference as big as the one observed due to sampling error is less than 0.05
define "statistical significance"
The extent to which a result deviates from the null hypothesis and is unlikely to have occurred by chance
What would it mean for the result of a hypothesis test to be statistically significant at the 0.01 level?
Same as before, the max. probability of committing a type I error is 0.01
How does a one-tailed hypothesis test differ from a two-tailed hypothesis test?
One-tailed tests specify directionality and test 'greater than/less than' alternative hypotheses
Two-tailed tests do not specify directionality and just test 'difference/not equal' alternative hypotheses
What effect does choosing a one or two-tailed hypothesis test have on your results?
One-tailed tests have a less conservative critical statistic that makes it easier to reject the null hypothesis
Two-tailed tests have a smaller critical statistic, making it harder to reject the null hypothesis but less chance of making a mistake
Suppose the scores of an IQ test are normally distributed with a mean of 100 and a standard deviation of 10. What percentage of students will score less than 93?
24.2% will score less than 93
Suppose the scores of an IQ test are normally distributed with a mean of 100 and a standard deviation of 10. What percent will score between 97 and 103?
23.6% will score between 97-103
Suppose the scores of an IQ test are normally distributed with a mean of 100 and a standard deviation of 10. How high might one need to score to be in the 95th percentile?
You would need to score 116.5 to be in the 95th percentile
Suppose I survey 144 University of Utah students about the number of alcoholic beverages each consumed on Saturday night. My sample yields a mean of 7 with a standard deviation of 2. Construct and interpret a 95 percent confidence interval for the true mean number of drinks consumed amongst the entire population of U of U students.
We’re 95% sure that the mean number of alcoholic beverages consumed for all University of Utah students is between 6.7 - 7.3
A sample of 900 Gotham City residents reveals that 49 percent of those surveyed support Batman’s quest for vigilante justice. Construct and interpret a 99 percent confidence interval for the true proportion of Gothamites who support Batman.
We’re 99% sure that the true proportion of Gothamites who support Batman falls between 44.7 - 53.2%
Suppose I know for a fact that the average American woman will give birth to 1.9 children over the course of her lifetime. An analysis of local fertility data gleaned from a sample of 121 women living in the greater Salt Lake area, however, reveals that the average female Utahan will give birth to 2.9 children with a standard deviation of 2. Test the hypothesis that women in Utah have, on average, more children than women in the rest of the country.
Null hypothesis: there is no difference in the mean number of children
Alternative hypothesis: the mean number of children born by Utah women is greater than the rest of the country.
Obtained statistic = 5.56
DOF = 120
Critical statistic = 1.645
Reject the null, alternative is true
Suppose I’m interested in studying differences in ACT scores between boys and girls. A sample of 100 boys yields a mean reading score of 20 with a standard deviation of 2. A sample of 100 girls, however, yields a mean reading score of 21 with a standard deviation of 3. Test the hypothesis that there is a difference in mean reading scores between boys and girls.
Null hypothesis: there is no difference in the mean reading scores between boys and girls
Alternative hypothesis: there is a difference in the mean reading scores between boys and girls
Obtained statistic = - 2.774
DOF = 198
Critical statistic = 1.96
Reject the null, alternative is true