Midterm 2 Flashcards
Population
Parameter
Statistic
Population
–entire group of interest
Parameter (P - Pop.)
–numerical fact about Population
(MU - ave. GPA of all BYU students)
Statistic (S = Sample)
–numerical fact about Sample (x bar = ave. GPA of 170 students in sample)
Sampling distribution of sample mean (x bar) is a theoretical prob. distribution
–it describes the distribution of: …
It describes the distributions of:
- -all sample means
- -from all possible random samples
- -of the same size
- -taken from same pop.
MEAN (x bar) = mean of sampling distribution of x bar
SD (x bar) = st. dev. of sampling distribution of x bar
relationship btwn sample size and st. dev.
can change n, but if do samples 500x - the mean all becomes SAME - but as N increases, the st. dev. decreases (less variation)
–variation in sample mean values tied to size of each sample - NOT the # of samples
Distribution of X bar for ALL possible SRS of size n from a pop. with mean MU and ST. D sigma
- -center: MEAN of sampling distribution of x bar is Mu (pop. mean)
- -Spread: st. dev. of sampling distribution of x bar is (sigma / radical n)
In real research…not realistic to estimate sampling distribution of sample mean by actually taking multiple random samples from same pop. - but there is a math. machine that shows
–math staticians have figured out how to predict what the sampling distribution will look like w/o actually repeating numerous times and having to close a sample each time
Central limit theorem
- -if you take a SRS of size n from any pop. then the shape of the sampling distribution of x bar is approx. normal
- -shape gets more normal as N increases
- n > 30, considered large
- -CLT allows us to use st. normal table to compute approx. probability associated with x bar
- -The sampling distribution of statistic (like a sample mean) often follows a normal distribution if sample sizes are large
center = Mu = pop. mean
spread = st. dev. = sigma / radical n
shape =
–case 1 - pop. is normal, shape distribution of x bar is normal
–case 2 - pop. is not normal - the shape of distribution of x bar is approx. normal when N is large (> 30) CLT
why do we care about the sampling distribution?
- -sampling distribution allows us to assess uncertainty of sample results
- -if we knew the spread of sampling distribution we could now how far our x bar might be from the true Pop. mean
if sampling distribution has a lot of variance - if you took another sample it is likely to get a very diff. result
- -about 95% of the time, the sample mean will be within (z * (sigma / radical n)) of pop. mean
- -this tells us how “close” the sample statistic should be to the pop. parameter
ex. what is probability that mean of n=75 will exceed $2700? Mu = 2600, st. dev. = 500
(2700 - 2600) / (500/ radical 75) = z = 1.73
- -on table pop. = .0418
- -have to do the sample st. dev. for the denominator
- -if normal than the sample also normal
- -so can compute prob. less than 32 - bc normal curve just like pop. - eve if small sample size bc pop. is normal
Fill in blanks:
the sample distribution of x bar gives —– from all possible samples of same size from same pop.
the sample distribution of x bar gives ALL X BAR VALUES from all possible samples of same size from same pop.
Suppose we take all possible samples of the same size from a population and for each sample, we compute x̄. The mean of these x̄ values will be exactly equal to the mean of the population (μ) from which the samples were taken - TRUE
Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the mean of the sampling distribution of x̄?
EQUAL TO 80
–must be equal to pop. mean bc large sample size
Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the standard deviation of the sampling distribution of x̄?
Less than 20
–large sample size = small st. dev.
What is the advantage of reporting the average of several measurements rather than the result of a single measurement?`
The average of several measurements is MORE LIKELY to be close to the true mean than the result of a single measurement
—NOT always = true mean…but more likely
Take on sample of size N - compute X bar for sample. as estimate (guess) of Mu
–use knowledge of sampling distribution of x bar in general to say something about uncertainty associated with this x bar of SAMPLE DISTRIBUTION
Center = mean of x bar
Spread = s of x bar = sigma / radical n
Shape = 1. pop. is normal = normal
2. large sample size >30 = normal
CHART
if talking about: 1. AN INDIVIDUAL (x) --pop. of ind. is normal? = YES use z = (x - Mu) / sigma then the normal table to find p =NO can't solve stop now
- A sample mean (x bar)
–Pop. of ind. is normal???
= YES
use z = (X bar - MU) / (Sigma / radical n)
–then normal table
=NO
check sample size
= > 30 CLT so use (z = (x bar - Mu) / ( sigma / radical n)
–then normal table
n <30 = nothing you can do
If all possible samples of size 80 are taken from a population instead of size 20, how would this change the mean and standard deviation of the sampling distribution of x̄?
The mean would stay the same and the standard deviation would decrease
GOOGLE STATS 121 EXAM 2 QUIZLET
True or False: We can never compute probabilities on x̄ when the population is skewed.
False
The following scenario applies to questions 3-5:
Suppose we have a normal population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the probability of getting an x̄ greater than 75?
.9938
–I did the pop. x bar formula and divided it by sigma over radical 100
statistical process control (jelly bean ex.)
series of interconnected steps in producing a product of service (ex. jellybeans)
–rec. raw ingredients - boil to syrup - add flavor - molded into shape - sugar coat - shell - glaze = branded = shipped
statistical Dogma for processes
all processes have NATURAL variation
–raw material, human performance, equipment performance, measurement
all processes susceptible to UNNATURAL variation
- -bad batch of raw material
- broken machine
Stats process control 2
use statistical paradigm to monitor process variables (inputs, outputs, etc.) over time to decide if variability consistent with natural variation
- -if consistent…continue process
- -if inconsistent, stop process - find cause of unnatural variation, fix problem, resume process
x bar control chart
in control process
out of control process
x bar control chart
–tool for monitoring variables of a process, alerting us when unnatural variation seems to have occurred
in control process
–process whose output exhibits only natural variation over time
out of control process
–process exhibits unnatural variation over time
Use central limit theorem to determine if process is in statistical control
- -ex. if jelly belly weight is normally distributed with mean MU = 1.1 grams and sigma = .1, then by CLT:
1. the sampling distribution of x bar is normal with mean 1.1 and st. dec. sigma / radical n
- “natural variation” of x bar is within 3 (sigma / radical n) of 1.1
constructing control chart for x bar
- draw horizontal centerline at Mu (pop. mean)
- draw horizontal control limits at
MU +/- 3* (sigma / radical n) - lot the means (x bar’s) from samples of size n against time
out of control signals
- one point above upper control limit or below lower control limit
- run of 9 POINTS IN a row on SAME SIDE of centerline
as soon as out of control signal is observed, STOP the process and look for a cause
What is the purpose of a statistical control chart?
To distinguish between natural and unnatural variation
inference testing
inference - drawing conclusions about a pop. (parameter) based on a sample (Statistic) with a measure of uncertainty
- -everything we’ve learned so far is to ensure valid inference
- -making generalizations about the population based on sample data
Pop. 1. producing data from pop.
- exploratory data analysis (data from sample)
- probability
- inference about pop.
Types of statistical inference 1. POINT ESTIMATION
- POINT ESTIMATION
–quantitative data ex.
based on sample of n = 47 policies, we estimate that the ave. premium is approx. $1800
–categorical data ex.
based on a. sample n = 144 households, we estimate that the proportion of infected bamboo is approx. 10.4%
Types of statistical inference 2. INTERVAL ESTIMATION
- INTERVAL ESTIMATION
–quant. data ex.
based on sample of n = 47 policies, we estimate that the ave. premium is btw $1700 and $1900
–categorical ex.
based on sample of n = 144 households, we estimate that the proportion of infected bamboo is btw 8.4% and 12.4%
Types of statistical inference 3. HYPOTHESIS TESTING
- HYPOTHESIS TESTING
–Quant. ex.
insurance agent believes that the ave. premium at her agency is $2500. Based on the sample n = 47 claims, we found that the ave. premium was $1800. this data, therefore, provides evidence that the ave. premium is less than $2500. That is, x bar = $1800 is an outcome that would rarely happen if the ave. was indeed $2500.
–categorical ex.
researchers believe that the proportion of infected bamboo is 2%. Based on a sample of n = 144 households, found that prop. was 10.4% This data provides evidence that the prop. is > 2%
Point estimation definition
–estimator and estimate
ESTIMATOR
- -general statistic that estimates the parameter
- -ex. the estimator of the pop. mean MU is the sample mean of x bar
ESTIMATE
- -a specific value of an estimator
- -ex. the ave. value of n = 47 is $1800
- the prop. of infected boards for n = 144 is 10.4%
- -the estimator of the pop. proportion p is the sample proportion p (with hat)
how good is the estimator x bar for MU?
A.) sampling needs to be done RANDOMLY
B.) sampling distribution of x bar tells us:
- on ave. x bar will give us the right answer - Calle this property UNBIASEDNESS
- As sample size n INC. the accuracy of x bar INC. (smaller st. dev)
Statistical inference chart
- CONFIDENCE INTERVAL point estimate +/- ME
- Test of significant res. fail to rej. at level alpha
both solve for:
- -conclusion about parameter
- -measure of uncertainty
LOOK AT CHART - CH. 17
four steps for confidence intervals
STATE
–Specify parameter of interest
PLAN
–choose procedure, level of confidence
SOLVE
–check conditions, carry out procedure
CONCLUDE
–interpret confidence interval
LOOK AT MY WRITING ASSIGNMENT
four steps for tests of significance
STATE
–Specify claims about parameters of interest
PLAN
–choose procedure, specify Ho, Ha, alpha
SOLVE
–check conditions, calc. test statistic and p-value
CONCLUDE
–compare p-value to alpha, interpret test results
What do we use to estimate μ?
M (aka median)
X BAR
A recent study claimed that half of all college students “drink to get drunk” at least once in a while. Believing that the true proportion is much lower, the College Alcohol Study interviews an SRS of 14,941 college students about their drinking habits and finds that 7,352 of them occasionally “drink to get drunk”. What type of statistical inference is this?
Hypothesis testing
Inferences for Mu (in unrealistically simple case)
- -conclusion about MU
- -gather data using SRS
- -Sigma known (rarely known for pop.)
- -sampling distribution of x bar is normal bc.
1. pop. is normal or
2. large sample, due to CLT
construct 95% confidence interval
–Plan to gather a random sample of 100 students. x bar estimates mea systolic blood pressure of all students
parameter: mean systolic blood pressure of all students
- -sample size: n = 100
- -st. dev. - sigma = 14.0 mm HG
how close should x bar be to Mu? informal distribution. then by 68,95,99.7 rule we cam say:
–the prob. that x bar is w/in 2 st. dev. (214/ radical 100 = 2.8) of Mu is .95
OR we are 95% confident that Mu is within 2(14/radical 100) of x bar or that interval x bar +/- 2* sigma/radical n contains MU
CAN’T SAY “PROBABILITY” WHEN TALKING ABOUT MU, So use “confident”
ex. n = 100, x bar = 123.4, sigma = 14 confidence interval can b written as (120.6, 126.2) OR (120.6 - 126.2) OR (120.6 to 126.2)
FORMULA FOR A 95% CONFIDENCE INTERVAL
X bar +/- 1.96* sigma/radical n
the 1.96 uses table of st. normal prob.and corresponds to the MIDDLE 95% of normal distribution
use confidence interval to fin z. and c% confidence int. for MU
–genreal formula for c% cond. int.
X bar +/- z* (sigma/ radical n)
z+ = confidence level you want
z is found at bottom of C TABLE
IF Sigma (pop. st. dev.)is UNKNOWN = one sample t confidence interval for means
- -We replace sigma with s
- -we replace z* with t*
bar +/- t* s/ radical n
t* = level of confidence you want
IF data gathered using SRS
- -Sigma unknown
- -normality of pop. distribution or large sample size
then:
sampling distribution of (x bar - mu) / (s / radical n) has a student’s t-distrib.
–WITH n-1 degrees of freedom DF
properties of t-distribution
- -symmetric, bell-shaped, mean = 0
- -the smaller the DF the larger the spread (bc more uncertainty due to s)
- -the larger the DF, the closer the t-distrib. to the standard normal
format of table of t-distrib.
- each t distribution is determined by its DF: df = n - 1
- if actual DF is not on table, use DF closest to actual without going over
- t* values are found in bod of table C
one sample t confidence interval outline
- STATE problem
- PLAN
- -Select procedure: one sample t CI for means
- -select confidence level
- -select parameter of interest in context - SOLVE
–collect and plot data
–calc. c bar and s
–check conditions (randomness and normality of pop distribution/ large sample size w/ no outliers
–calc. confidence interval using formula:
bar +/- t* (S / radical n)
4. CONCLUDE interpret CI in context including --confidence level --parameter of interest --calculated interval
EX. One sample t conf. int. for MU
- STATE: students at large church activity took survey of 121 married students to estimate 2/ 90% confidence how long these students had dates on ave. before getting engaged
- PLAN: use “one sample t CI for MU
- -confidence level:90%
- -parameter of interest: mean # of months married students dated before engagement - SOLVE: collected SRS of 121 married students
s = 10, x bar = 9.34
–conditions: random (SRS)
–normal or large sample size, n = 121
(1st CHECK IF CONDITIONS MET..IF ARE CAN CONT. AND FIND CI)
9.24 +/- (1.66) * (10/radical 121) =
= (7.83, 10.85)
- CONCLUDE:
- -we are 90% confident (NOT PROBABILITY) that the interval (7.8 mo, 10.8 mo) contains the TRUE MEAN # of mo. married students at tis activity dated before getting engaged
definition of confident
in repeated sampling, the confidence level is the percentage of confidence interval produces by procedure that actually contain the value MU, or the success rate of the procedure
ex. 95 of all possible 95% conf. interval estimates for MU actually contain the value of MU
- -the confidence is based on the PROCEDURE, not the INTERVAL
True or False: Increasing the sample size will lead to a wider margin of error.
false
Suppose a sample of size 250 was taken instead of size 100. How will the margin of error change?
–margin of error would DECREASE
True or False: Increasing the confidence level will lead to a wider margin of error.
TRUE
Which one of the following is the correct representation of the margin of error when sigma is unknown?
t* sigma/ radical n
MARGIN OF ERROR DOES NOT INCLUDE THE SAMPLE MEAN
just the second part of equation after =/-
What is the name of the quantity (s/ radical n?)
standard error of x-bar
breaking up the formula
x bar +/- z* sigma/radical n
- -x bar = point estimator
- -z* = confidence multiplier
- -radical n = st. dev. of pt. estimator
- -everything RIGHT of +/- is MARGIN OF ERROR (m)
as sample size INC, margin of error DEC.
-AS N INC, width of confidence. interval DEC.
–confidence level INC, margin of error INC
also can be written as x bar +/- m.
margin of error
- margin of error (m) controls the width of the interval
- as sample size INC, m and width DEC
- as conf. INC, m and width INC
m = (z*sigma)/ radical n
ex. sigma = .25, m must be no larger than .05 with 99% confidence
- -how many times should PH be measured?
n = ((z* sigma)/m)^2 = 165.89 = 166
–ALWAYS ROUND UP TO NEXT INTEGER (even if 165.1 — 166).
a 90% confidence int. is (.21, 2.45) what is the margin of error?
(.21 + 2.45)/ 2 = 1.33
- 33 - .21 = 1.12
- 45 - 1.33 = 1.12
M = 1.12!!!
–Margin of error is the distance from lower bound to mean and upper bound to mean
standard error of x bar
= s / radical n
–not just S!!!!!
S/ radical n = estimates the st. dev. of the sampling distribution of x bar
In confidence interval estimation, we use a confidence interval to estimate ———-
the value of a population parameter
The t-distribution with 6 degrees of freedom has ____________________ the standard Normal distribution.
t disturb. has the same center but is more spread out than st. normal distribution
The purpose of a confidence interval is to provide
plausible values that a parameter could take
–NOT a measure of the confidence we can have in our sample results representing the population
hypothesis testing
- draw conclusion about PARAMETER using STATISTIC
- WITH a measure of uncertainty
- confident interval = estimates a parameter
- test of significant = hypothesis testing = to assess a claim about a parameter’’
“significantly reduced” = mean score was TRULY reduced
- -can prove it is NOT true but can’t prove it is
- -we failed to disprove a claim, can’ prove it is true but can prove life its not
ALWAYS assume claim that researchers think is NOT true
- -approach called PROOF by CONTRADICTION (can’t disprove so can argue it is true)
- -researchers think claim is FALSE so they assume it was TRUE
4 elements o test of significance
- CLAIM 1: (Ho) - Null hypothesis - always be explained in problem as ___ = ____
CLAIM 2: (Ha) - alternate hypotheses - always be ___ , not = ___ - OUTCOME - standardized outcome that measures how far the outcome diverges from CLAIM 1 (p-value)
- -outcome represented by standardized stat. called a “Test stat” (p-value) - ASSESSMENT OF EVIDENCE
- -how likely is it to get this outcome if claim 1 is true?
- -p-value - prob. will get something in the tail - CONCLUSION
- -an outcome that would rarely happen if claim is true is good evidence that Claim 1 is not true, hence we believe claim 2 is true
stats about significance test
ALPHA meaning
related measure of uncertainty is alpha which represents the prob. of falsely rejecting claim 1 or Ho : alpha defines what is rare or unlikely outcome
a test with >/< in Ha is a ONE-SIDED TEST
test with NOT = is a two-sided test
test statistic
number that summarizes data for a test of significant
- -companies an estimate of parameter from sample data with value of parameter given in null hypothesis
- -measures how far sample data verge from Ha
- -large va are not consistent with Ho: evidence against Ho
- -used to find prob. of obtaining sample data IF Ho were true
- -ex. of test statistic
t = bar - Mo /(s / radical n)
meaning of p-value
a # btw o and 1
0 <= P-val. <= 1
-the prob. of getting test stat as extreme or more extreme than observed if Ho were true
–measure of strength of agreement btw observed test stat and Ho
–measures evidence against HO
STRENGTH OF EVIDENCE AGAINST Ho - PVAL IS THE PROBABILITY THAT H0 IS TRUE!! LOW PROB. MEANS SHOULD REJECT
Pval close to 0 = good evidence H0 is not true = evidence for Ha
Pval - .5
p val. = 1
no evidence that Ho is not true
= not evidence for Ha
pre-specified cut-off for P-val.
artificial but imp. - sharp boundary btw rejection and non-rejection regions for p-value
–if p-val. <= alpha - diff. is statistically significant, reject Ho and conclude it as false
ALPHA = ASSUME .05 UNLESS SPECIFIED IN PROBLEM
Alpha and P-val.
P-val. <= alpha: REJECT Ho
- -declare observed diff. statistically significant
- -conclusion: believe Ha
- -diff. btw claimed parameter value and calculated statistic likely real, not chance
- -stat tests do not address issue of importance (“practical significance”
P-0val > alpha: do not reject Ho
- -do not declare statistacally sign.
- -insufficient evidence to believe Ha
- -don’t accept Ho but fail to reject Ho
- -fail to rejectHo means the diff. could be due to chance, not real
Alpha definition
risk of false positive (reject null when actually true)
–risk should generally be small(
T/F the NULL hypothesis is the claim that the researcher ants to prove
FALSE
–Trying to prove the alternate hypothesis
one-sample t test examples
realistic case:
- -gather data using SRS
- -sigma unknown
- -pop. distributions. approx normal (Single peak, no excessively long tails)
- if sample does not have extreme outliers or skewness, can assume normality or pop.
- as n INC. skewness and non-normality less worrisome
replace sigma with s
x bar - Mu / (s / radical n)
st. dev. = radical (sum of (x - x bar)^2 / n-1)
if sigma unknown, random and normal/;are sample size
then: sampling distribution. of x bar - Mu / s/ radical n has a t-distrib. with n-1 DF
t = (x bar - mu) / (s / radical n)
95% conf. interval = x bar +/- s / radical n
false positive and false negative
false positive: rejecting null hypothesis when shouldn’t have
false negative - failing to reject null hypotheses when you shouldn’t have
TYPE 1 error: reject Ho when it is true: false positive - pronounce guilty when they are innocent
TYPE 2 error: fail to reject Ho when it is false - false negative - pronounce innocent when guilty
type 1 is more serious - put innocent person behind bars WORSE than letting guilty person roam free (type 2)
alpha and beta
alpha = level of significance
- -probability of type 1 error
- -probability reject Ho when it is true
BETA
- -probability of type 2 error
- -probability fail to reject Ho when it is false
POWER = probability (reject Ho when it is false) 1 = beta --power = .99 = 99% prob. of rejecting null hypothesis when it is actually false (want to be high!!)
LEARN THE GRAPHS!!!
N and power
- -alpha
- -beta
- -power
- -effect size
N INC. the sample distribution. gets tighter and narrower
larger sample size always INC. power
alpha = prob. type 1 error Beta = prob. type 2 error
power = prob. of rejecting Ho when it is false (with beta)
effect size = diff. btw Ha and Ho
(EX. mu = 28, x bar = 24, effect size = 4)
which is harder…detect diff. with sample size 10 or 100?
10 is harder - more data = easier
art is my bff
Type 1 error
Alpha
Reject Ho
TRUE
Type 2 error
Beta
Fail to reject Ho
False Ho
what influences power?
- effect size - # change in Ho and Ha
- variability in measurements (sigma = out pop. st. dev. - inc. var = dec. power, dec. sigma = inc. power)
- chosen significance level (alpha)
- sample size (N INC = power INC (bc dec. st. dev.)
significance depends on sample size
t = x bar - Mu / (s / radical n)
significance depends on:
x-bar mine Mo:
–size of observed effect(numerator to test stat.)
–measures how far the sample mean deviates from the hypothesized Mo
–the “LARGER” the observe effect, the smaller the p-val. (which tests the prob. Ho should be accepted)
–size of sample n
–s/radical n - measures how much random variation we expect
–larger sample size = smaller p-val.
–sample size may be too small to detect significance
–sample size may be so large results are always significant
when sample size is large check for practical importance
results are declared STATISTICALLY SIGNIFICANT when P-val. <= alpha
Results practically important when observed effect (numerator of test stat.) is large or imp. enough to mater
practical importance is not same as statistical significant
–practical im. determined by common sense
large samples - unimportant diff. can be significantly significant
small sample - important diff. may not be statistically signifiant
–always ask whether stat. significant effect is “large enough to matter”
practical importance
declare statistically significant if P-val. <= alpha
- -consider practical. imp. when observed effect large - practical value
- -observed effect: numerator of west stat (x bar - Mo)
1st. check for STAT. SIGNIFICANCE
2. d = Check PRACTICAL IMP.
P-val for 2 sided tests
2 * p-val. of one sided test - 2 sided req. stronger evidence than one-sided
- inspector significance vs. 2, inspector interval
- conduct test of significance at alpha = .05
–Ho: mean egg weight = 50g
–Ha = mean egg weight does not = 50 g
compare p-val. with alpha and draw conclusions
–find t = (x bar - Mo)/ (s/radical n) then look at DF for that 2 and make an interval - inspector interval
- -construct 95% CI for Mu
- -see if interval includes or excludes value 50
- -if CI includes 50: data provides support for Ho
- -if CI excludes 50: data provides evidence to reject Ho
x bar +/ t* s/ radical n
True or False: The null hypothesis is the claim that the researcher wants to prove.
FALSE
True or False: p-value gives the probability that the null hypothesis is true.
FALSE
–A p-value is a conditional probability—given the null hypothesis is true, it’s the probability of getting a test statistic as extreme or more extreme than the calculated test statistic
For many years, “working full-time” has meant working 40 hours per week. Nowadays it seems that corporate employers expect their employees to work more than this amount. A researcher decides to investigate this hypothesis. The null hypothesis states that the average time full-time corporate employees work per week is 40 hours. The alternative hypothesis states that the average time full-time corporate employees work per week is more than 40 hours. To substantiate his claim, the researcher randomly selected 250 corporate employees and finds that they work an average of 47 hours per week with a standard deviation of 3.2 hours. In order to assess the evidence, what do we need to ask?
How likely it is that, in a sample of 250, we will find that the mean number of hours per week full-time corporate employees work is as high as 47 if the true mean is 40?
Suppose the p-value is 0.0367. What is the correct interpretation of this p-value?
Assuming the null hypothesis is true, there is a 0.0367 probability of obtaining a sample statistic as extreme or more extreme than what we calculated.
Reject the null hypothesis. The true mean hemoglobin level of all children in Jordan is less than 12 g/dl.
Suppose the p-value for this test is 0.0734. What are the appropriate conclusions to make when α = 0.05?
Fail to reject the null hypothesis. There is insufficient evidence to conclude that the mean hemoglobin level of all children in Jordan is less than 12 g/dl.
NOT - Fail to reject the null hypothesis. The mean hemoglobin level of all children in Jordan is equal to 12 g/dl - can’t accept!! But know Ha not true
If we set α = 0.05, what can we do to increase power?
Increase sample size.
