Checking Conditions
In the following statement which of the 3 conditions (random, independence, & normal) are not met:
Joey is a basketball sports better. Joey knows that the number of points any player makes on a specific team is normally distributed with a mean of 10 and a standard deviation of 4. Joey creates a sample of his 4 favorite players and his 4 least favorite players.
Joey's sample size is 8 and basketball teams are not 80 players therefore the independence condition is not met.
Additionally Joey is not selecting a random sample.
Normal condition is met since Joey samples from a normal distribution.
A researcher found that if people are asked to rate how cute a penguin is on a scale of 1 to 10 the average response is 8.0 with a standard deviation of 0.8. What is the probability that a SRS of size 10 yields a sample mean less than 7? In other words,
P(\bar{x}<7.0)=?
A convenience sample is an example of a(n) ________ estimator. While a SRS is an example of a(n) _______
biased estimator;
unbiased estimator.
Draw a boxplot if your 5 number summary is:
1,4,6,11,15
I'll judge the drawing.
In the following statement which of the 3 conditions (random, independence, & normal) are not met:
In a voting district of 10,000 individuals 5400 approve a certain bill. A SRS of size 20 is taken and individuals are polled on whether they approve or disapprove of the bill.
Random: met
Independence: met
Normal: not met 20*.46 = 9.2 < 10
If p = 0.3 and n = 40, what is,
\sigma_{\hat{p}}?
=\sqrt{\frac{0.3*0.7}{40}}=\sqrt{0.00525}=0.072
A _______ is a number that describes some characteristic of the population, while a ________ is a number that describes some characteristic of a sample.
parameter; statistic
Arrange this data into a stemplot.
{18,18,17,20,22,23,21,25,25,24,23}
1 | 7,8,8
2 | 0,1,2,3,3,4
2 | 5,5
The proportion of blue eyed individuals in a certain state is 0.11. Suppose that you plan on making a SRS of size n and that the state has a population of N = one million. What is the minimum amount of people you can sample such that the sampling distribution is approximately normal?
0.11*n >= 10 implies that n has to be at least 91.
A researcher found that if people are asked to rate how cute a penguin is on a scale of 1 to 10 the average response is 8.0 with a standard deviation of 0.8. What is the probability that someone rate the cuteness of a penguin less than 7?
NormCdf(-infinity, 7, 8,0.8) = 0.106
True or False: Sampling variability is the idea that taking samples from the same population will typically yield a variety of different sample statistics (such as different means, different sample proportions, etc.)
True
A random variable X has a standard deviation of 1.2, and a random variable Y has a standard deviation of 9. Let Z = X - Y. What is the standard deviation of Z?
\sigma_{Z}=\sqrt{\sigma_{X}^2+\sigma_{Y}^2}=\sqrt{1.44+81} \approx 9.08
Suppose you are sampling from a normal distribution with a mean of 12 and a standard deviation of 2. Suppose you want
\sigma_{\bar{x}}
to be less than 1. What is the minimum sample size required?
You need a sample size of 5 at least, since
\frac{\sigma}{\sqrt{5}}=\frac{2}{\sqrt{5}} < 1
In a voting district of 10,000 individuals 5400 approve a certain bill. A lobbyist wants to push against this bill; to do so he takes a SRS of size 30. What is the probability that the lobbyist finds a sample proportion of approval or less than or equal to 0.49?
There is a 29.1% chance that the lobbyist will find such a sample proportion.
P(\hat{p}<0.49) = \text{NormCdf}(-\infty,0.49,0.54,\sqrt{\frac{0.54*0.46}{30}})
=0.291
True or False: According to the Central Limit Theorem, a sample size of any size will result in a normally distributed sample mean.
False: n has to be greater than or equal to 30.
In a least squares regression line, the slope b, is equal to r times what?
\frac{s_{y}}{s_{x}}
You take a SRS of size n from some distribution (not necessarily normal). The population you are sampling is of size N = 100. Is it possible to meet all 3 conditions with any value of n? Explain why.
No, because n must be less than or equal to 10 (independence condition) but must be larger than or equal to 30 to meet the normal condition.
In a certain district the demographics of the religious believes are 33% Catholic, 42% Christian, 10% Jewish, 12% Muslim, and 3% other. What is the probability that in a SRS of 31 individuals, 12 or more individuals will be Catholic?
NormCdf(12/31, infinity, 0.33, 0.084) = 0.249
True of False: The Central Limit Theorem states that as your sample size increases the sample mean gets closer and closer to the population mean.
False: This is the Law of Large Numbers we briefly mentioned in Ch. 5 when we discussed empirical probabilities approaching theoretical probabilities.
The probability of winning a certain grand prize from a lotto ticket is 0.001. Jerry plays the lotto daily (1 ticket per day). We expect Jerry to win the grand prize for the first time on his nth attempt, where n = ?
n = 1000, since this is a geometric distribution scenario the expected first success happens on the 1/p trials.