Exam 3
An investigator wishes to determine whether, for a sample of drug addicts, the mean score on the depression scale of a personality test differs from a score of 60, which, according to the test documentation, represents the mean score of the general population.
H0: u = 60 H1: u /= 60
Sampling distribution
a complete distribution (set) of a particular statistic (M, s, s^2) of size n taken from a population of size N
The mean of all sample means usually equals the value of a particular sample mean.
False. Because of chance, most sample means tend to be either larger or smaller than the mean of all sample means.
The mean of all sample means always equals the value of a particular sample mean.
False. It always equals the value of the population mean.
The standard error of the mean increases in value with larger sample sizes.
False. It decreases in value with larger sample sizes.
The central limit theorem ensures that the shape of the sampling distribution of the mean equals the shape of the population.
False. It ensures that the shape of the sampling distribution approximates a normal curve, regardless of the shape of the population (which remains intact).
The standard error of the mean measures variability in a particular sample.
False. It measures variability among sample means.
The central limit theorem states that, regardless of sample size, the shape of the sampling distribution of the mean is normal.
False. It requires that the shape be sufficiently large-usually between 25 and 100.
The central limit theorem states that, with sufficiently large sample sizes, the shape of the population is normal.
False. The shape of the population remains the same regardless of sample size.
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? This particular interval describes the population mean about 95% of the time.
False. This particular interval either describes the one true population mean or fails to describe the one true population mean.
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? About 95% of all subjects scored between 507 and 527.
False. We can be 95% confident that the mean for all subjects will be between 507 and 527.
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? The true population mean definitely is between 507 and 527.
False. We can be reasonably confident-but not absolutely confident-that the true population mean lies between 507 and 527.
There are two possible states of the world (in the population)
H0 = true - "no actual treatment effect" H0 = false - "real treatment effect"
To increase rainfall, extensive cloud-seeding experiments are to be conducted, and the results are to be compared with a baseline figure of 0.54 inch of rainfall (for comparable periods when cloud seeding was not done).
H0: u < 0.54 H1: u > 0.54
When untreated during their lifetimes, cancer-susceptible mice have an average life span of 134 days. To determine the effects of a potentially life-prolonging (and cancer-retarding) drug, the average life span is determined for a group of mice that receives this drug.
H0: u < 134 H1: u > 134
In hypothesis testing, there are two possible decisions
reject H0 retain H0
Type I error
rejecting H0 when, in reality, it is true
CI over population mean
sample is out/over performing
CI under population mean
sample is underperforming
H0: X bar = 241 H1: X bar does not equal 241
sample means (rather than population means) appear in H0 and H1
Sample with replacement
select someone from population, test as part of sample, return to population
Intersection of events
set of all points in both A and B
Union of events
set of all points in either A or B
Level of confidence
size of MOEs and CIs 95% confidence interval (p = .95) vs. 99% confidence interval (p = .99) corresponds to an area under normal curve alpha level is a = 1-p
Point estimates
statistical estimates of parameter -some are unbiased (e.g. - x) -some are biased (e.g. - S, S^2, SS) single values from a sample used to represent a parameter how close is sample mean to the parameter? "dance of the means" and the "mean heap" -variability in the values of means from randomly selected samples -sample means pile up into the "mean heap"
Independent events
the occurrence of one event has no effect on the probability that the other event will occur
Probability
the proportion or fraction of times that a particular event is likely to occur
Confidence interval (CI)
twice the MOE range around a sample mean that has a certain likelihood of including an unknown population parameter range around a sample mean of plausible values for u u in CI, sample performing consistently with population
Non-directional test (two-tailed)
uA /= uB
Directional test (one-tailed)
uA > uB or uA < uB
Event (A, B, C...)
unique outcome in a random experiment
Z-test
used to compare a single sample mean to a known population mean assesses whether a sample mean differs from the population mean/"norm"
Assume the same probabilities as in the previous question. For a randomly selected family with three children, what's the probability of neither three boys nor three girls?
(1-0.016) + (1-0.016) =
Assuming that people are equally likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what's the probability of both husband and wife being born during December?
(1/12)(1/12) = (1/144)
Assuming that people are equally likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what's the probability of the husband being born during January and the wife being born during February?
(1/12)(1/12) = (1/144)
The probability of a boy being born equals .50, or 1/2, as does the probability of a girl being born. For a randomly selected family with two children, what's the probability of two boys, that is, a boy and a boy?
(1/2)(1/2) = 1/4
The probability of a boy being born equals .50, or 1/2, as does the probability of a girl being born. For a randomly selected family with two children, what's the probability of two girls?
(1/2)(1/2) = 1/4
Assume the same probabilities as in the previous question. For a randomly selected family with three children, what's the probability of three boys?
(1/2)(1/2)(1/2) = 0.125
Assume the same probabilities as in the previous question. For a randomly selected family with three children, what's the probability of three girls?
(1/2)(1/2)(1/2) = 0.125
The probability of a boy being born equals .50, or 1/2, as does the probability of a girl being born. For a randomly selected family with two children, what's the probability of either two boys or two girls?
(1/4) + (1/4) = 2/4 = 1/2
Assuming that people are equally likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what's the probability of both husband and wife being born during the spring (April or May)?
(2/12)(2/12) = (4/144)
Assume the same probabilities as in the previous question. For a randomly selected family with three children, what's the probability of either three boys or three girls?
(3/2) + (3/2) = 0.016
Given a one-tailed test, lower tail critical with a = .01, and (a) z = -2.34 (b) z = -5.13 (c) z = 4.04
(a) Reject H0 at the .01 level of significance because z = -2.34 is more negative than -2.33. (b) Reject H0 at the .01 level of significance because z = -5.13 is more negative than -2.33. (c) Retain H0 at the .01 level of significance because z = 4.04 is less negative than -2.33.
Given a one-tailed test, upper tail critical with a = .05, and (d) z = 2.00 (e) z = -1.80 (f) z = 1.61
(d) Reject H0 at the .05 level of significance because z = 2.00 is more positive than 1.65. (e) Retain H0 at the .05 level of significance because z = -1.80 is less positive than 1.65. (f) Retain H0 at the .05 level of significance because z = 1.61 is less positive than 1.65.
Hypothesis testing
1. set up competing hypotheses 2. gather data 3. run statistics and determine outcome 4. make decisions about hypotheses based on outcome
Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during June?
1/12
Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during any month other than June?
11/12
Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during either May or June?
2/12 = 1/6
Indicate whether each of the following statements is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that... it is impossible to get 10 cards from the same suit (for example, 10 hearts).
False. Although unlikely, 10 hearts could appear in a random sample of 10 cards.
Indicate whether each of the following statements is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that... the random sample of 10 cards accurately represents the important features of the whole deck.
False. Just by chance, a random sample of 10 cards fails to represent the important features of the whole deck. Randomness describes the selection process-that is, the conditions under which the sample is taken-and not the particular pattern of observations in the sample. You can't predict anything about the unique pattern of observations in the sample.
Public health statistics indicate, we will assume, that American males gain an average of 23 lbs during the 20-year period after age 40. An ambitious weight-reduction program, spanning 20 years, is being tested with a sample of 40-year old men.
H0: u > 23 H1: u < 23
Indicate Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively. All U.S. presidents; all registered Republicans
No. All U.S. presidents aren't a subset of all registered Republicans.
Indicate Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively. Citizens of Wyoming; citizens of New York
No. Citizens of Wyoming aren't a subset of citizens of New York.
Should H0 be retained or rejected, given a hypothesis test with critical z scores of +/- 1.96 and z = -2.51.
Reject H0 at the .05 level of significance because z = -2.51.
Should H0 be retained or rejected, given a hypothesis test with critical z scores of +/- 1.96 and z = 013.
Retain H0 at the .05 level of significance because z = 1.74.
Should H0 be retained or rejected, given a hypothesis test with critical z scores of +/- 1.96 and z = 1.74.
Retain H0 at the .05 level of significance because z = 1.74.
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? In practice, we never really know whether the interval from 507 to 527 is true or false.
True
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? The interval fro 507 to 527 refers to possible values of the population mean for all students who undergo special training.
True
Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95% confidence interval of 507 to 527. Are the following interpretations true or false? We can be reasonably confident that the population mean is between 507 and 527.
True
Indicate whether each of the following statements is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that... any outcome, however unlikely, is possible.
True
Indicate whether each of the following statements is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that... each card in the deck has an equal chance of being selected.
True
The central limit theorem applies to the shape of the sampling distribution-not to the shape of the population and not to the shape of the sample.
True.
The mean of all sample means equals 100 if, in fact, the population mean equals 100.
True.
The mean of all sample means is interchangeable with the population mean.
True.
The standard error of the mean equals 5, given that q=40 and n=64.
True.
The standard error of the mean roughly measures the average amount by which sample means deviate from the population mean.
True.
Indicate Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively. 20 lab rats in an experiment; all lab rats, similar to those used, that could undergo the same experiment
Yes
Indicate Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively. Students in the last row; students in the class
Yes
Indicate Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively. Two tosses of a coin; all possible tosses of a coin
Yes
Random sampling
a selection process that guarantees all potential observations in the population have an equal chance of being selected hopes to make sample accurate occurs if, at each stage of sampling, the selection process guarantees that all potential observations in the population have an equal chance of being included in the sample describes a selection process, the conditions under which the sample is taken and not the particular pattern of observations in the sample
Null hypothesis
a statistical hypothesis that usually asserts that nothing special is happening with respect to some characteristic of the underlying population
Trial
act of performing a random experiment once
Addition rule
add together the separate probabilities of several mutually exclusive events to find the probability that any one of these events will occur works only when none of the events can occur together a. P(A U B) = P(A or B) = P(A) + P(B) b. mutually exclusive events c. P(A U B) = P(A or B) = P(A) + P(B) - P(A)P(B)
Pr(A or B) = Pr(A) + Pr(B)
addition rule for mutually exclusive events Pr = probability of the event in parentheses A and B are mutually exclusive events
Sample space (S)
all possible outcomes in a random experiment
Sampling distribution of the mean (M)
all sampling means (M) of sample size (n) randomly selected from a population i. standard deviation of the sampling distribution of the mean ii. average difference between u and M of size n iii. n error (sampling error) between u and M
Population
any complete set of observations (or potential observations) "attitudes toward abortion of currently enrolled students at Bucknell University" "SAT critical reading scores of currently enrolled students at Rutgers University"
Random assignment
any procedure or operation with uncertain outcome
Sample
any subset of observations from a population smaller group that accurately represents the population
H0: u = 155 H1: u does not equal 160
different numbers appear in H0 and H1
Mutually exclusive events
events that cannot occur together
Type II error
failing to reject H0 when in reality, it is false
In hypothesis testing:
i. state a hypothesis that assumed to be true ii. collect data and calculate an "observed result" iii. determine probability of observed result, which is, P(observed result/hypothesis = true) iv. if the probability is extremely low, the result is extremely rare given the hypothesis is true
One-tailed or directional test
if there is a concern that the true population mean differs from the hypothesized population mean only in a particular direction must retain H0 regardless of how far the observed z deviates from the hypothesized population mean in the direction of "no concern"
Central limit theorem
in a population with u and q, a sampling distribution for sample size n has uM = u sampling distribution of mean approximates a normal distributions as n approaches infinity
Margin of error (MOE)
largest likely (most probable) estimation error (sampling error) between u and M largest likely difference we should expect between a sample mean (M) and a population mean (u) u can be outside MOE, it's just unlikely to be
Pr(A and B) = [Pr(A)][Pr(B)]
multiplication rule for independent event
Multiplication rule
multiply together the separate probabilities of several independent events to find the probability that these events will occur together a. P(A and B) = P(A)P(B/A)
Alternate hypothesis
opposite of null hypothesis
Sample without replacement
possible to be reselected
