Sampling distribution/CLT
You ask 80 people whether chocolate glazed is their favorite donut or not; 20 say yes. State whether the quantity we can calculate from the given information is a statistic or parameter. Calculate and express the value being sought using the correct notation.
Statistic. P-hat = 20/80 = 0.25
You suspect that less than 25% of Americans have heard of your favorite podcast. If you wanted to find out whether this is true or not, would you use a hypothesis test or a confidence interval? And would it be for proportions or means?
Hypothesis test, proportions
The population distribution is the distribution of cases in the population. A sample distribution is the distribution of cases in the sample. The sampling distribution is the distribution of _______?
sample statistics (e.g. sample means or sample proportions)
"If you take many different samples of the same size from the same population and then construct a 95% confidence interval for each of them, about 95% of these will contain mu.” Is this true or false?
True (think of the simulations we did in the last asynchronous video)
True or False: The mean of the sampling distribution of sample means for samples of size n = 15 will be the same as the mean of the sampling distribution for samples of size n = 100.
True (by the first part of the Central Limit Theorem these are both equal to mu)
You want to know if less than half of all college students have their own credit card. The null is that half of them have their own credit card (p = 0.5). The alternative is that less than half have their own credit card (p < 0.5). Your alpha is 0.05 and calculate your test statistic to be -1.5. What is your rejection decision and your plain English conclusion?
Your p-value is 0.0668, which is greater than alpha. So, you fail to reject. There is not enough evidence to conclude that less than half of all college students have their own credit card
The mean of the SAT is 1060 and SAT scores are normally distributed.
Which of the following is the most UNLIKELY:
A. Drawing a single student with a score of 1300 or above
B. Drawing a sample of 5 students with an X-bar of 1300 or above
C. Drawing a sample of 30 students with an X-bar of 1300 or above
C. Drawing a sample of 30 students with an X-bar of 1300 or above
I am interested in the proportion of Wellesley students who prefer to study in the library. If I take two different samples of the same size from the same population and then construct a 95% confidence interval for each of them, are the two intervals necessarily the same length?
Not necessarily! Your interval length is determined, in part, by your SE. Your SE is determined, in part, by your p-hat. So if you get different p-hat's in your samples then the two intervals you make will be of different lengths.
Suppose you work for the Red Cross. Tomorrow you will collect blood from 32,000 donors. Based on demand, you need at least 1,850 of the donors to be O-negative. About 6% of people have blood type O-negative. What’s the probability that you will have enough O-negative donors show up tomorrow?
z = (0.0578 - 0.06)/0.001328 = -1.66
The area above this is 95.15% so that's your answer. (This question requires use of the CLT and Z-scores).
You have found a coin that you suspect is rigged in favor of heads. You do a hypothesis test in which the null is that the coin is fair; the alternative is that it is unfair in favor of heads. You flip it 100 times and get 60 heads.
In a single sentence, interpret this p-value.
If the coin was fair, the probability of it giving you a result this unusual (60 or more heads) would be 0.0228.