Z-scores
In the summer, country A has mean temperature 85 degrees Fahrenheit with SD 3 degrees. Also in the summer, country B has mean temperature 78 degrees Fahrenheit with SD 8 degrees. Which is the most unusual: It is 90 degrees in country A, it is 90 degrees in country B, or these are equally likely.
The Z-score for country A is z = (90 - 85)/3 = 1.67. The Z-score for country B is z = (90 - 78)/8 = 1.5. It would be more unusual for country A to have a 90 degree day, since they have the larger Z-score in magnitude.
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
Blood pressure in healthy individuals is normally distributed. The mean is 120 mm Hg with a standard deviation of 15 mm HG. Ali has a blood pressure of 131 mm HG. What percentile is she in?
z = (131 - 120)/15 = 0.73
76.73% of healthy people have a Z-score below 0.73.
Ali is then in the 76th percentile.
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)
IQ score is normally distributed in adults. What percent of adults have an IQ that is at least two SD's above the mean?
2.5% (use the empirical rule of thumb)
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
Blood pressure in healthy individuals is normally distributed. The mean is 120 mm Hg with a standard deviation of 15 mm HG. What’s the probability of a healthy individual having a blood pressure between 110 and 130?
0.4972
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).
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.