STATS 10 Final

Let A be the event that a person is left handed and B be the event that the person is over 30 year old. Write symbolically: The probability that: a person is a lefty who is over 30.

P(A AND B)

De Morgan's Laws

(A intersect B)^C = A^C union B^C (A union B)^C= A^C intersect B ^C

When is H0 rejected? P value approach

-If the p-value is less than .01, H0 is rejected. The results are highly significant. -If the p-value is between .01 and .05, H0 is rejected. The results are statistically significant. -If the p-value is between .05 and .10, H0 is usually not rejected. The results are only tending toward statistical significance.

binomial experiment 5 Characteristics

1. The experiment consists of n identical results. 2. Each trial results in one of two outcomes: one outcome is called a success, S, and the other a failure, F. 3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to (1 - p) q. 4. The trials are independent. 5. We are interested in x, the number of successes observed during the n trials, for x 0, 1, 2, ..., n.

A FAIR COIN HAS COME UP "HEADS" 10 TIMES IN A ROW. THE PROBABILITY THAT THE COIN WILL COME UP HEADS ON THE NEXT FLIP IS 1. less than 50%, since "tails" are due to come up. 2. 50% 3. greater than 50%, since it appears that we are in a streak of "heads." 4. It cannot be determined.

2. They are independent from each other so it will always be 50%

Ch.7: Q4) It is generally believed that electrical problems affect about 14% of new cars. An automobile mechanic conducts diagnostic tests on 128 new cars on the lot. What is the sampling standard deviation (Sampling Error)? 1. 0.347 2. 0.001 3. 0.031 4. Cannot determine from information given.

3

A certain population is approximately normal. We want to estimate its mean, so we will collect a sample. Which should be true if we use a large sample rather than a small one? I. The distribution of our sample data will be approximately normal. II. The sampling distribution of the sample means will be approximately normal. III.The variability of the sample means will be smaller. 1. I only 2. II only 3. III only 4. II and III 5. I, II, and III

5

Which of the following is not an assumption or a condition that needs to be checked for the one-proportion z-test? 1. Independence 2. Randomization 3. 10% Condition 4. Success/Failure Condition 5. Nearly Normal Condition

5

What does standard error tell us?

A standard error is the standard deviation of the sampling distribution of a statistic. Standard error is a statistical term that measures the accuracy with which a sample represents a population. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error.

stratified sample

A stratified sample is one obtained by separating the population into homogeneous, non-overlapping groups called strata, and then obtaining a simple random sample from each stratum.

Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome

Law of Total Probability

Assuming that A1, A2,..., An are partisans then, P(B) = P(B given A1) + P (B given A2)..+ P(B given An)

55% of the songs on John's iPod are Classical. Everyday, John creates a new random playlist of 25 songs. He does this for 100 days. Each day he writes down the proportion of the songs that are Classical. Which of the following is true? Select one: A. The distribution of the 100 sample proportions of classical songs is skewed to the left. B. the distribution of 100 sample proportions of classical songs is nearly normal. C. The distribution of the 100 sample proportions of classical songs is skewed to the right. D. The distribution of the sample proportions would be nearly normal if there were many more than 100 playlists.

B. np>10 and n(1-p)>10, so the CLT says the distribution will be approximately normal.

Hyp testing Classical method VS P value

Both you calculate Z the same way. For classical, you find Z subalpha and based on that reject or accept null. For P value approach, find the P value. If right tail for ex, look at the area to the right of the calculated z-score. If that area is less than the alpha given in the problem, then you reject null

Suppose that in a certain country the distribution of age at which females get married follows the normal distribution with μ = 27 years and σ = 2 years. If you randomly pick a female that is younger than 30% of the female population at the time of the marriage, how old will she be? Select one: A. 30 years old B. 26 years old C. 28 years old Correct D. we need the median to decide

C

Hyp testing for small samples

Check whether np0 and n(1-p0) is greater than or equal to 10, where p0 is the proportion stated in the null hypothesis. If the that is true, then we may do the same methods as with the top portion. If NOT true then do the following: Compute the P-value. For right-tailed tests, the P-value is the probability of obtaining x or more successes. For left-tailed tests, the P- value is the probability of obtaining x or fewer successes. Use the binomial formula. The P-value is always computed with the proportion given in the null hypothesis. Step 4: If the P-value is less than the level of significance, , we reject the null hypothesis.

The number of standard errors to move away from the mean of the sampling distribution to correspond to the specified level of confidence is called the.......

Critical Value

if P is not statistically significant do we fail to reject or reject null hypotheses?

Fail to reject null

Hyp testing Type 2 error

Failing to reject when the null is wrong OR failing to reject null when Alternative is right Beta

Confidence interval

Formula given in Formula sheet. Things to know: if confidence is 95% then in order to find Z sub Alpha, you need to do: 1-.95= .05----> .05/2----> .025. This .025 os what is to the right of the z value, however the z score table gives you what is to the left so if its the table from econ 41, then you do 1-.025 and look for that area under the table. the z score for this example should be 1.96

We have calculated a 95% confidence interval and would prefer for our next confidence interval to have a smaller margin of error without losing any confidence. In order to do this, we can I. change the z∗ value to a smaller number. II. take a larger sample. III. take a smaller sample.

II

what does Alpha mean in terms of type 1/2 error?

It is the probability of type 1 error

Null Hypothesis Vs Alternative

Null = H0 which is the hypothesis of no change Alternative= H1 which is the one which you are trying to find evidence to support

Bayes Rule

P(A given B)= [P(B given A)* P(A)] / P(B)

Let A be the event that a person is left handed and B be the event that the person is over 30 year old. Write symbolically: The probability that: a person over 30 years old is a righty.

P(AC|B)

Let A be the event that a person is left handed and B be the event that the person is over 30 year old. Write symbolically: The probability that: a left handed person will be over 30.

P(B|A)

Indepedent Events

P(a and b)= P(a)*P(b) P(a given b)= P(a)

The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that at least one of 500 randomly selected 60 year old females will die during the course of the year?

P(at least one dies) = 1 - P(none die) = 1 - P(all survive) = 1 - 0.99186500 = 0.9832

According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog?

P(do not own a dog) = 1 - P(own a dog) = 1 - 0.316 = 0.684

Shape of binomial distribution with p=.1, P=.5 and p=.9 all with N= 10 what happens as N increases?

P=.1 ---> right skewed P=.5----> symmetric P=.9----> left skewed As N increases, it becomes more symmetric and bell shaped!!!! Which allows us to use the empirical rule

if P is statistically significant do we fail to reject or reject null hypotheses?

Reject

sampling error

SQRT [(p(1-p))/n)] where P is mean (MAKE SURE TO DEFFERENTIATE THIS WAY OF CALCULATION SD AND THE PRE MIDTERM WAY)

standard error formula

Standard deviation/sqrt(n) OR Sqrt[p(1-p)/n]

Probabilities: Expection/mean

Summation of x*Probability X=x

Based on a random sample of 164 fourteen-year-old girls in the U.S., a researcher finds that a 95% confidence interval for the height all 14-year-old girls in the U.S. is (63.3 inches to 64.7 inches). If the researcher wants a 99% confidence interval, what will change? Select one: a. The margin of error will be larger. b. The standard error will be smaller. c. The margin of error will be smaller. d. The standard error will be larger.

The more confident we want to be, the larger our ME needs to be, making the interval wider. A.

An economist believes that Americans will spend less money on holiday shopping this year than last year. Last year, 85% of all Americans bought a gift for the Winter holiday season. The economist publishes results of a survey of Americans that, he says, shows that a lower proportion of Americans are spending money on gifts this year. He shows this with a hypothesis test, and says that he used "a significance level of 1%." What does this mean?

The probability that he will conclude that a lower proportion are shopping when, in fact, the proportion is 85%, is 1%. (REMEMBER alpha= probability of type 1 error) The probability that he will get a proportion more extreme than the one he saw, assuming that in fact 85% of Americans are shopping this year, is 1%.

CLT

The sampling distribution model of the sample mean (or proportion) from a random sample is approximately Normal for large n, regardless of the distribution of the population, as long as the observations are independent.

The variability we expect to see from one random sample to another is called ___________

The sampling error

Confidence vs precision. T or F: To be more confident, we wind up being less precise.

True because to be more confident you need to INCREASE your interval which means less precision. IF you want to be more precise, that means a shorter interval which means less confidence that your true parameter is there

Mutually exclusive

Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. (that is there is no overlap between two events in a ven diagram). P(A B) = 0 P(A U B) = P(A) + P(B)

Two Sample Design Vs Paid design

Two-sample design: compares responses for two independent groups. Paired design: a pair of response values is recorded for each unit.

HYp testing step 5: rejecting or accepting

Two-tailed: Z0<-Zsubalpha/2 or if Z0>Zsub alpha/2 , then reject null hypothesis Left tail: Z0<Zsubalpha, then rejectNull Right Tail: Z0>Zsubalpha, then reject null P value approach: If the P-value <alpha, reject the null hypothesis.

The level of significance, , is the probability of making a Type 1 OR 2 error?

Type 1 error As prob of type 1 error Increases, prob of type 2 error decreases

Normal distribution notation

X ~ N (m , s) Z ~ N(0,1) <--- standard normal

Z score

Z= X-mean/ SD

How to find confidence Interval. Z underscore Alpha/2 if its 95% confidence

alpha= 1- Critical Value Alpha= 1-.95 = .05 alpha/2=.025 ----> Zunderscore .025 Remember, when its in this form, its telling you that .025 is to the right of that z value. Therefore, to find the z value, you must do 1-.025= .975 and look for that value since the table gives you the are to the left. IF the table is the one that is 0 at the mean, then do .5-.025

Theoretical probabilities

are long run relative frequencies based on theory. P(Heads) = 0.5

Empirical probabilities

are short run relative frequencies base on an experiment. A coin was tossed 50 times and landed on heads 22 times. The empirical probability is 22/50 = 0.44.

math Symbol: >=

at least, no less than , greater than or equal to

A 2009 study by Pew Research Foundation found that 37% of a sample of 1022 Americans believe that global warming is real. To find a 95% confidence interval for the proportion of all Americans that believe this, they calculate: 0.37 plus-or-minus 1.96 * sqrt(0.37*0.63/1022). This leads to a confidence interval of (0.34 to 0.40). Which of the following is the margin of error? Select one: a. 0.06 b. 0.03 c. 1.96 d. sqrt(0.37*0.63/1022)

b

A survey was conducted at a movie theater to determine movie-goers' preference for different popcorn. The results of the survey show that Brand A was preferred by 65% of the people with a margin of error of plus or minus 3 percent. What does the statement "plus or minus three percent" mean? Select one: A. The true proportion of the population who preferred Brand A popcorn could be determined is 3 percent more of the population was surveyed. B. It would be unlikely to get the observed population of 65% unless the actual percentage of people who preferred Brand A is between 62 and 68 percent. Correct C. Three percent of the population surveyed will change their minds. D. Three percent of the surveys are not accurate. E. Three percent of the population was surveyed.

b

Assume that resting systolic blood pressure for healthy women under the age of 35 has a mean of 120 and a standard deviation of 9. Also assume that the distribution of these women's systolic blood pressures is unimodal and symmetric. If we assume that blood pressure is normally distributed, about 16% of healthy women... Select one: A. have resting systolic blood pressure between 102 and 111 B. have resting systolic blood pressure above 129 Correct C. have resting systolic blood pressure below 102 D. have resting systolic blood pressure above 138

b

Which best describes a sampling distribution model of a statistic? Select one: a. It is the probability that the sampling statistic equals the parameter of interest. b. It is a distribution of all the statistics calculated from all possible samples of the same size. c. It is the probability distribution of all values that are contained in all possible sample of the same size. Incorrect d. It is a histogram of of the one of the samples.

b. It is a distribution of all the statistics calculated from all possible samples of the same size.

The Federalist Papers are a collection of essays written anonymously by some of the Founding Fathers after the American Revolution to try and convince the 13 states to form a federal government. Although we know who the authors are, we aren't sure who wrote which essay. Essay 28 may have been written by Alexander Hamilton or by James Madison. A historian has noticed that in Hamilton's writings, he started 20% of his sentences with a preposition, while Madison rarely started a sentence with a preposition. Of the 200 sentences analysed from Essay 28, the historian finds that 18 sentences start with a preposition. He conducts a test of the hypothesis H0: p = 0.20 against Ha: p not = 0.20. What is his p-value, and what conclusion can he draw from it? Select one: a. p = 0.12. He should not reject the null hypothesis, that Alexander Hamilton wrote the essay. b. p = 0.00. He should reject the null hypothesis, and conclude that James Madison wrote the essay. c. p = 0.00. He should not reject the null hypothesis, that Alexander Hamilton wrote the essay. d. p = 0.012. He should reject the null hypothesis, that Alexander Hamilton wrote the essay. e. p = 0.00. He should reject the null hypothesis, and conclude that Alexander Hamilton wrote the essay. Incorrect

b. p = 0.00. He should reject the null hypothesis, and conclude that James Madison wrote the essay.

We have discussed that the standard deviation of the distribution of sample percentages or SE (p^) is calculated by taking the square root of P*(1-P) divided by N, where P is the proportion in the population and N is sample size. What does SE (p^) show? Select one: a. It shows the variance of the p^ distribution. b. It shows the distribution of p^ for the single sample that the researcher draws from the population. c. It shows the standard deviation of the P distribution. d. It shows the standard deviation of the p^ distribution.

d. It shows the standard deviation of the p^ distribution

Records at a large university (not UCLA, of course) indicate that 20% of all freshmen are placed on academic probation at the end of the first semester. A random sample of 100 freshmen found that 25% of them were placed on probation. The results of the sample: Select one: a. are biased since an increase of 5% could not happen without injecting bias into the sample. b. are surprising since it indicates that 5% more of these freshmen were placed on probation than expected Incorrect c. are surprising since SAT scores have increased over the past years. d. are surprising since the standard deviation of the sampling distribution is 0.4%. e. are not surprising since the standard deviation of the sampling distribution is 4%.

e. are not surprising since the standard deviation of the sampling distribution is 4%.

z sub alpha

is the z-score such that the area under the standard normal curve to the right of zα is α.

math Symbol: <

less than, fewer than

Math Symbol: >

more than, greater than

Math Symbol: <=

no more than, at most, less than or equal to

Hyp testing: Point estimate P

p=x/n where x is the number of individuals in the sample with the specified characteristic and n is the sample size.

Hyp Testing Type 1 error

rejecting Null Hyp when its true Alpha alpha = probability of making this error

non-sampling errors ex

• Incomplete populationFrame for selecting the sample • Non-response • Poor Interviewer • Incorrect Data Entry • Poorly worded question • Open versus Closed Question • Ordering of Questions Sampling error is the error that results in using sampling to estimate information regarding a population