Stats Test 1

6 step investigation

-1) ask research question -2) design study and collect data -3) explore data -4) draw inferences beyond data -5) formulate conclusions -6) look back and ahead

estimating population proportion

p hat +/- Square root p hat (1-p hat)/n

statistics for samples

- sample mean= x bar - sample stedev= s

parameters for population

-population mean= mu -population stedev= sigma

theory based interval for mu

-required quantitative variable to have symmetric distribution OR -have at least 20 observations and distribution not strongly skewed

clt for sample mean conditions

-sample size big enough (n>20) and not too skewed OR -distribution of sample is bell shaped if n<20

3s strategy

1) statistic- compute stat from observed sample data 2) stimulate- identify "by-chance-alone" explanation for data 3) strength of evidence- consider if value of observed stat is unlikely to occur when chance model is true

d

A t-distribution is shaped like a normal distribution but is: A. A bit skewed in one direction or the other. B. A bit taller in the middle than a normal distribution. C. Not quite as spread out with fewer observations in the tails than a normal distribution. D. A bit more spread out and more observations in the tails than a normal distribution.

b

According to enrollment data in 2016, 15% of undergraduates at a private college were international students. Your statistics professor wondered if the proportion of international students at the college changed after Donald Trump became the President of the United States. She surveys 100 of her students and finds that 10 are international students. Determine whether the alternative hypothesis would be one-sided or two-sided. Question options: a) One-sided b) Two-sided

d

After you conduct a coin-flipping simulation, a graph of the _______ will be centered very close to 0.50. Choose from (A)-(D). A. Process probability B. Sample size C. Number of heads D. Proportion of heads

b

As the sample size increases, the standard deviation of the null distribution of the sample proportions will also increase. Question options: a) True b) False

a

Identify which of the following is the null hypothesis. Question options: a) The long-run proportion of city residents over the age of 20 who fit the medical definition of obese is equal to 0.396. b) The long-run proportion of city residents who fit the medical definition of obese is equal to 0.396. c) The sample proportion of city residents who fit the medical definition of obese is equal to 0.396. d) The sample proportion of 80 city residents who fit the medical definition of obese is equal to 0.275.

d

If you are testing the hypotheses H0: π = 0.50 and Ha: π ≠ 0.50, have a sample proportion of 0.60 and get a p-value of 0.321, what can you say about a 95% confidence interval constructed using the same data? A. The 95% confidence interval will definitely contain 0.321. B. The 95% confidence interval will definitely not contain 0.321. C. The 95% confidence interval will definitely not contain 0.60. D. The 95% confidence interval will definitely contain 0.50. E. The 95% confidence interval will definitely not contain 0.50.

statistic

number summarizing results in a sample

quantitative

numerical values where arithmetic operations make sense

b

Reconsider Exercises 1.2.9 and 1.2.10 about your friend shooting free throws. You run this test and find a p-value of 0.1150. Which of A-E is the best way to state the conclusion? A. Because the p-value is large, there is strong evidence that your friend is a 75% free-throw shooter in the long run. B. Because the p-value is not small enough, there is not strong evidence that your friend is less than a 75% free-throw shooter in the long run. C. Because the p-value is small, there is strong evidence that your friend is less than a 75% free-throw shooter in the long run. D. Because the p-value is small, there is strong evidence that your friend is a 75% free-throw shooter in the long run. E. Because the p-value is large, there is strong evidence that your friend is a 65% free-throw shooter in the long run.

standard error

SD/sqrt(n)

b

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 40 free throws and she makes 26 (or 65%) of them. Which of A-E is an appropriate way to set up the hypotheses, in symbols, for this test? A. H0: = 0.75, Ha: < 0.75 B. H0: π = 0.75, Ha: π < 0.75 C. H0: = 0.65, Ha: < 0.65 D. H0: π = 0.65, Ha: π < 0.65 E. H0: π < 0.75, Ha: π = 0.75

c

Suppose that you test the hypotheses H0: p=0.50 and HA: p<0.5. You collect some data and get a sample proportion of 0.45. From this you compute a standardized statistic and p-value. If your sample proportion had been 0.30 instead, then your Question options: a) standardized statistic and p-value would both increase. b) standardized statistic would decrease and p-value would increase. c) standardized statistic would increase and p-value would decrease. d) standardized statistic and p-value would both decrease.

c

Suppose you are conducting a test of significance to try to determine whether your cat, Hope, will go to the correct object (out of two) when it is pointed to, just like Harley the dog did in the Exploration 1.1. You test Hope 100 times and finds she goes to the correct object in 70 of those 100 trails. In this scenario, what is the parameter? A. The 100 trials B. Whether or not Hope goes to the correct object C. The long-term proportion (probability) that Hope will go to the correct object D. The proportion of the 100 trials that Hope goes to the correct object

at null hypothesis

Suppose you are testing the hypotheses H0: π = 0.25 and Ha: π < 0.25 and the observed statistic, is equal to 0.30 with a sample size of 100. a. If you are using a proportion as your statistic, where do you expect your null distribution to be centered?

c

Suppose you are testing the hypothesis H0: p=0.50 versus HA: p>0.5. You get a sample proportion of 0.61 and find that your p-value is 0.04. Now suppose you redid your study with a sample size twice as a big and still find a sample proportion of 0.61. How will your new p-value change? Question options: a) It will get bigger. b) It will stay the same. c) It will get smaller.

d

Suppose you are using simulation to determine p-values. How will a two-sided p-value compare to a one-sided p-value, assuming the one-sided p-value is less than 0.5? Question options: a) The two-sided p-value will be exactly twice as large as the one-sided p-value. b) The two-sided p-value will be about the same as the one-sided p-value. c) The two-sided p-value will be about half as large as the one-sided p-value. d) The two-sided p-value will be close to twice as large as the one-sided p-value.

d

Suppose your hypotheses are H0: p=0.25 and HA: p<0.25. In this context, which of the standardized statistics below would provide the strongest evidence against the null hypothesis and in favor of the alternative hypothesis? Question options: a) z = -1 b) z = 3 c) z = 0 d) z = -1.8

a

The larger the p-value, the closer a standardized statistic gets to zero. Question options: a) True b) False

a

The p-value of a test of significance is: A. The probability, assuming the null hypothesis is true, that we would get a result at least as extreme as the one that was actually observed B. The probability, assuming the alternative hypothesis is true, that we would get a result at least as extreme as the one that was actually observed C. The probability the null hypothesis is true D. The probability the alternative hypothesis is true

a

The simulation (flipping coins or using the applet) done to develop the distribution we use to find our p-values assumes which hypothesis is true? A. Null hypothesis B. Alternative hypothesis C. Both hypotheses D. Neither hypothesis

c

When we get a p-value that is very large, we may conclude that: A. The null hypothesis has been proven to be true. B. There is strong evidence for the alternative hypothesis. C. The null hypothesis is plausible. D. The alternative hypothesis has been proven to be false.

b

When we get a p-value that is very small, we may conclude that: Question options: a) The null hypothesis is plausible. b) There is strong evidence for the alternative hypothesis. c) The alternative hypothesis has been proven to be false. d) The null hypothesis has been proven to be true.

c

Which hypothesis testing method is appropriate for this study? Question options: a) Neither method is appropriate b) Simulation-based test c) Either method is appropriate d) Theory-based test

c

Which of A-C is the most important reason that a simulation analysis would repeat the coin-flipping process many times? A. To see whether the distribution of sample proportions follows a normal, bell-shaped curve B. To see whether the distribution of sample proportions is centered at 0.50 C. To see how much variability results in the distribution of sample proportions

d

Which of A-D is NOT true about theory-based confidence intervals for a population proportion? A. They should only be used when you have at least 10 successes and 10 failures in your sample data. B. The process used to construct the interval relies on a normal distribution. C. They can be calculated using different confidence levels. D. For a given sample proportion, sample size, and confidence level, different intervals can be obtained because of their random nature.

2 sided test

____ provides less evidence against the null hypothesis since it is twice the size of a one sided test

sampling frame

a list of individuals from whom the sample is drawn -ex: class registrar

margin of error

accounts for sample to sample variability in sample proportion

bell shaped sampling distribution

approx 95% of statistics in sampling distribution fall w/ in 2 stedevs of mean

increases

as the confidence interval increases, does the width of the interval also increase/decrease?

no

can a sampling frame claim to represent segments in population with sample if it does not include every member of population?

larger

can a smaller/larger sample size tolerate more skewness?

binary variable

categorical variable with only 2 outcomes

categorical

category designations (cannot perform arithmetic operations)

width of interval

confidence interval 2- confidence interval 1

data table

convenient way to store and represent data values

tall of distribution

observations that fall more than 2/3 standard deviations away from the mean

confidence interval

observed sample +/- margin of error

clt for sampling from large finate population

distribution of samples proportions from repeated random sample will be approx normal is at least 10 successes and 10 fails in each sample

sampling distribution

distribution of statistic for all possible samples of size randomly selected from same population

narrower

does a large sample lead to the width of the 95% interval to become narrower or wider?

larger

does larger/smaller sample size increase the strength of evidence against the null hypothesis?

decreases

does standard deviation increase/decrease with a larger sample size?

null hypothesis

does the p-value assume the alternative or null hypothesis is true until evidence is provided?

population

entire collection of observational units

standard error

estimate of stedev of statistic

take margin of error and add to lower CI

how do you find p-hat/ the proportion that responded in a sample?

clt for sample means

implies distribution of sample mean is approx normal when sample size is large -n>20 is large

increased

increased/decreased probability of success in null hypothesis and moved closer to sample proportion the p-value becomes larger?

observational units

individual entities on which data are recorded

yes

is the prediction of theory-based valid with 10 success and 10 fails?

farther away

is there more evidence against the null hypothesis when the observed stat is farther away or closer to the mean of the null distribution?

less

is there more/less sample to sample variability in sample mean than individual sample

parameter

long-run numerical property of process for a random process

probability

long-run proportion of times an event would occur if random process where repeated under a very large number of times with identical conditions

confidence level

measure of how confident we are about our interval estimate of the parameter

shape of t distribution

mound shape curve centered at zero, more spread out than normal distribution (wider)

convince sample

non-random sample (voluntary responses)

sample size

number of observational units in a sample

value of parameter

plausible if 2-sided p-value for testing that parameter is high/larger than the level of significance -aka not enough evidence against null value, so null value is plausible

parameter value

plausible if it's not more than 2 stedevs from the observed value of the stat -95% CI= all values of pi that are w/in 2 stedevs of p hat (only valid w/ bell shaped symmetric distribution= 2sd method)

probability distribution

predictable pattern of results when many trials are performed

p-value

probability of obtaining a value of the statistic at least as extreme as observed stat when null hypothesis is true

variables of interest

recorded characteristics of observational units

null hypothesis

represents "by-chance-alone" explanation -chance model chosen to reflect this hypothesis

alternative hypothesis

represents "there is an effect" explanation that contradicts the null hypothesis

statistically significant

result if it is unlikely to occur just by random chance -if observed results appears to be consistent with chance model, chance model is plausible/believable

sample

set of observational units on which we collect data

pi

symbol that means parameter is a probability

p hat

symbol that stands for sample proportion

true

t/f, w/ simple random sample. mean of sampling distribution of sample mean will equal the population mean

left skew

tail on left, mean<median

right skew

tail on right, mean>median

standardize a statistic

to compute the distance of the observed stat from the hypothesized mean of a null distribution and divide it by the standard deviation of the null distribution - z= statistic- mean of null distribution/ standard deviation of null distribution

standard deviation

typical distance between data values and mean distribution

significance level

value used as criteria for deciding how small p-value needs to be to provide convincing evidence to reject the null hypothesis

smaller p-value

values of statistic are even further from hypothesized parameter result in ______ and stronger evidence against null hypothesis

population proportion

what are confidence intervals used to estimate the value of?

parameter

what are the null and alternative hypothesis only about?

rows

what are the observational units in a data table?

columns

what are the variables in a data table?

simulation

what can be used to estimate a probability?

distribution of variable

what describes the pattern of value/category outcomes?

null distribution

what distribution has these characteristics? -often follow (not always) bell-shaped curves -centered at null hypothesis value for pi -their variability is influenced by sample size

null value

what does a confidence interval defiantly contain?

sample proportion

what is definitely contained in a 2sd/theory based confidence interval?

margin of error

what is half the width of the interval (CI2-CI1/2) called?

simulation

what is the 2sd value based on

random sample

what is this from a large population distribution of sample mean? -center around population mean (mu) -have stedev smaller than population

confidence interval

what is used to estimate long-run proportion (probability)/population proportion?

z stat

what tells us that p hat is ____ of a standard deviation from the mean?

when sampling normal distribution

when is the 2sd method for 95% confidence interval approporiate?

null hypothesis

where is the null distribution centered at?

simple random sample

which type of sample ensures every sample size is equally likely to be picked?

formula for theory based CI for mu

x bar +/- multiplier (SD/sqr n)