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Unformatted text preview: Chapter 6 Sections 6.1, 6.2, 6.3 Statistical inference is the next topic we will cover in this course. We have
been preparing for this by 1) describing and analyzing data (graphs and
plots, descriptive statistics) 2)discussing the ways to find and/0r generate
data (studies, samples, experiments) and 3) we have studied sampling
distributions. We are now ready for statistical inference. The purpose of statistical inference is to draw conclusions about population
parameters based on data which came from a random sample or from a
randomized experiment. Data (statistics) are used to infer population parameters. We will learn in this chapter the two most prominent types of statistical
inference, o conﬁdence intervals for estimating the value of a population mean
and 0 tests of signiﬁcance which weigh the evidence for a claim concerning
the population mean. Section 6.1 Estimating the population mean, u, with a stated conﬁdence: A confidence interval for the population mean, M, includes a point estimate
and a margin of error. 0 The point estimate is a single statistic calculated from a random
sample of units. For example, x, the sample mean, is a point
estimate of ,u , the population mean. However sample means ﬂuctuate, so we need to adjust our point estimate by adding and
subtracting a margin of error, thus creating a confidence interval. The following general formula may be used when the population sigma is
knOWn: % Confidence Interval: Xi margin of error , where *0" 7—,"; Margin of error— — 5‘ Lecture 7, Chapter 6
Page 1 For the Normal Distribution the following values for Z* apply:
For a 90% Confidence Interval Z* = 1.645
For a 95% Confidence Interval Z* = 1.960 For a 99% Confidence Interval Z* = 2.576 Example: You want to estimate the population mean SAT Math score for the high
school seniors in California. You give a test to a simple random sample of
500 high school seniors in CA. The mean score for your sample, Y 2461.
The population standard deviation is known to be 6 = 100. For the SAT Math scores, a 95 % confidence interval for [1. would be : 461 + /  1.960 *100/ \l (500) which works out to:
(452.2, 469.8 ) This Confidence Interval was calculated using a procedure which gives a
“correct” interval (Contains the Population Mean), 95% of the times it is
used. Another example:
1. Suppose we wish to estimate ,u, the population mean driving time between Lafayette and Indianapolis. We select a SRS of n = 25 drivers. The observed sample mean is x1 = 1.10 hours. Let’s assume that we
know the population standard deviation of X is 0' = 0.5 hours. A 95 % confidence interval for ,Lt would be: 1.10 + / — 1.96”< 0.5/V (25) which works out to: (0.904, 1.296) Again, the procedure gives a “correct” interval
95% of the times it is used. Lecture 7, Chapter 6
Page 2 Suppose we selected another sample of 25 driving times and obtained an E
= 1.00 hours. If we calculate a 95% confidence interval for ,u based on }2 we get (0.804, 1.196) which is a different interval estimate.
Which interval is correct? We don’t know. i If we repeatedly selected SRS of 25 drivers, and for each SRS, we calculated
a 95% confidence interval for ,u, the population mean, in the longrun, 95% of the intervals will contain the true value of ,u . They will all be different, but 95 % of them will include the true mean and Will
therefore be considered correct. And the other 5% will not include the
true mean . We have no way to know whether a given Confidence
Interval is “correct” or “incorrect”. Confidence level vs Width of Confidence Interval:
Suppose X, Bob’s golf scores, have a normal distribution with unknown
population mean but we believe the population standard deviation 0' = 3. A SRS of n=16 units is selected and a sample mean of E = 77 is observed.
a. Calculate a 90% confidence interval for ,u. Use Z* = 1.645 We CI: Pom :2 77 3: l.é#5(%) : 77 2!: 1.23 1:. (75377 3 78.2 3)
b. Calculate a 95 % confidence interval for ,u. Use Z* = 1.960 £257. :17: Fara z: 77 5.: manta} == 774:. W7
:2 (75.537 WW7) 0. Calculate a99% confidence interval for ,u. Use Z2: 2.576
am CI 53:3? .44 :1 2?“? 1*: 2.5% m“ a 77 :3: W3 £75.57 79'93) As you can see from these calculati ns, raising th confidence interval
requires a larger Z* value, which increases the margin of error and produces
a wider confidence interval. There is a trade—off between the precision of
our estimate and the confidence we have in the result. Higher conﬁdence
level requires a wider interval. Lecture 7, Chapter 6
Page 3 The margin of error also depends on sample size. A larger sample size will
result in a smaller margin of error. In fact, quadrupling the sample size
will cut the margin of error in half. Calculating the sample size for a desired margin of error: The confidence interval for a population mean will have a specified margin
of error m when the sample size is 2
>1:
n=z0
m 1. You are planning a survey of starting salaries for recent liberal arts major
graduates from your college. From a pilot study you estimate that the
population standard deviation is about $8000. What sample size do you
need to develop a 95% confidence interval with a‘ margin of error of $500? Example: 2. Always round a sample size number with any decimals up to the
next whole number. Never drop the decimals and round down. Some Cautions: o The above formulas do not correct the data for any unknown bias.
Consequently, if the data are biased, then ANY inferences based on those
data are also biased. This includes biases arising from nonresponse,
undercoverage , response error or hidden bias in experiments. 0 Because the sample mean is not resistant, confidence intervals are not
resistant to outliers. 0 When the population being sampled is not normally distributed, the
sample size needs to be at least 30 in order to have the sample mean be
normally distributed. This is the Central Limit Theorem. Always plot
the data to check normality. Lecture 7, Chapter 6
Page 4 0 Typically we do not know the population standard deviation, 0'. When 0' is not known we will use the t procedures which will be introduced in
Chapter 7. Interpretation Of A Conﬁdence Interval: Any value in a confidence interval is considered a possible value for [1,
including the end points. Any value not included in the confidence
interval is considered an unlikely value for u. Section 6.2 TESTS OF SIGNIFICANCE (HYPOTHESIS TESTING)
The second type of statistical inference is a significance test which assesses evidence provided by data regarding some claim about the population mean. Based on a random sample from the population, we want to determine if a
the population mean has changed upward, downward, or in either direction.
Because the null hypotheSes represents the established or accepted mean
value, we want to use the data to determine, statistically, if we can reject the
null hypothesis in favor of the alternative hypothesis. The four steps for a Test of Significance/Hypothesis Tests: Step 1. State the Null and Alternative Hypothesis:
Null Hypothesis H 0 : The statement being tested in a statistical test is called the null hypothesis. The test is designed to assess the strength of the
evidence against the null hypothesis. Usually the null hypothesis is a
statement of “no effect” or “no difference” or “status quo”. H 0 :1“ 2 [“0
Alternative Hypothesis H a : The claim about the population mean that we are trying to find evidence for. Choose one of the following hypotheses.
H a :,u > #0 one side right H a :,u < #0 one side left
Ha :,u¢,uo two side Step 2. Find the test statistic:
If [“0 is the value of the population mean [1 specified by the null Lecture 7, Chapter 6
Page 5 ‘ai hypothesis, the onesample z statistic is
= x ‘ﬂ0
0'/ x/ﬁ Z Step 3. Calculate the pvalue. For one sided left tests: the P Value = P(Z 5 z), the area in the left tail.
For one side right tests: the P Value = P(Z 2 z), the area in the right tail.
For two sided tests: the P Value = 2P(Z Z z) , the area in the right and
left tail Step 4. State conclusions in terms of the problem. The value of or defines
how much evidence we require to reject Ho. Then, compare the pvalue to
the Qt level. This is usually stated in the problem. If pvalue = or < a, then reject H 0 . Strong evidence exists against Ho. If pvalue > a, then fail to reject H 0. Insufficient evidence exists..... Generally the value chosen for or is one of the following three: 0 OL = .01 strictest burden of proof of these three values.
0 OL = .05
0 or = .10 easiest burden of proof of these three values. Your conclusion should be in this form: If the p—value =< 0t we say that we have sufficient evidence to reject the null
hypotheSis in favor of the alternate hypothesis, using the words of the
original problem. If the pvalue > 0t we say that we do not have sufficient evidence to reject the
null hypothesis, using the words of the original problem. Even though H a is what we hope or believe to be true, our test gives
evidence for or against H 0 only. We never prove H 0 true; we can only state whether we have enough '
evidence to reject H 0 (which is evidence in favor of H a , but not proof that H z is true.) or that we don’t have enough evidence to reject H 0 . Lecture 7, Chapter 6
Page 6 .5? ’. Example 1, A one sided hypothesis test: Bob’s golf scores are historically normally distributed with ,u = 77 strokes
and o = 3 strokes. Bob has recently made two “improvements” to his game,
and he thinks his scores should be lower. Bob has played 9 rounds since these improvements. His scores are:
77 73 74 78 78 75 75 74 71 Sample Mean=75 :2. Does this data provide sufficient evidence to conclude that Bob’s population
mean is reduced, ie, [1. < 77? Null Hypothesis: Ho: u = 77 Status Quo 0r Established Norm
Alternate Hypothesis: Ha: u < 77 Improvement Since we know that Bob’s golf scores are normally distributed, the sampling
distribution of the sample mean of 9 rounds must also be normally
distributed. The standard deviation of 3c— is 0} =3/J9 = 1 stroke. The logic of the hypothesis test:
If H 0 is true, then X ~ N (77,1) If H 0 is false and H a is true, then 7 ~ N (,u, 1)
for some value ,u < 77 . 0 Values of X close to 77 would tend to support H 0 and values that
are much lower than 77 would provide evidence against H 0 and in favor of Ha. 0 From the sample of Bob’s last 9 scores we get a sample mean of
X = 75. Can we conclude that we should reject H 0 in favor of H a ? 0 We need to calculate the Pvalue. Assuming that H0 is true, we
calculate the probability P(Y< 75) = P[Z < 75177] = P(Z < —2.00) 20.0228 0 The calculation says that if H0 is true, the probability that Xbar would
be < 75 solely due to random chance is .0228, or 2.28%. 0 For this example let us use a = .05. Lecture 7, Chapter 6
Page 7 he X 0 Because the probability of obtaining a sample E < 75 is less than a
we would reject H 0 and conclude it is more likely that u < 77. Example 2.
What if Bob had only obtained the first 5 scores. In this case, we get a sample mean of )_C = 76 and 07¢ = 3/ «[5 21.342. Then the P Value 76 — 77
1.342
meaning that an Xbar value of 76 or lower could occur by chance alone 22.66% of the time when H0 is true. This is not strong evidence that the
population mean has changed. POT— < 76) = P [Z < ]= 0.2266 which is much greater than 0L, 0 So we would fail to reject HO . Example 3, A two sided hypothesis test: Bob has a driver’s license that gives his weight as 190 pounds. Bob’s
license is coming up for renewal. Let’s test Whether Bob’s weight is
different from 190 pounds using a test of significance. Let’s assume that
Bob’s weight is approximately normally distributed with a population
standard deviation of 3 pounds. Bob’s last four weekly weights are: 193 194 192 191 Sample Mean = 192.5: 3(— We ask if this data provides sufficient evidence to say that Bob’s weight has
changed. (Two side hypothesis wording because direction is not implied) Null Hypothesis: Ho: [1. = 190 No change
Alternate Hypothesis: Ha: [1. 75 190 Changed. Two side hypothesis Again, the parameter value specified in the null hypothesis usually
represents no change, or the status quo. The suspected change in the
parameter value is stated by the alternative hypothesis. Calculate Z: (192.5—190) / (3/sqrt(4)) = 1.67 Lecture 7, Chapter 6
Page 8 Calculate P Value: Tail probability = .0475 For two side tests we must double the tail probability.
P Value = 2 ( .0475) = .0950 which is the probability in both tails. Reach a conclusion using OL = .05: Since the P Value is greater than 0t, we
have insufﬁcient evidence to reject H0. We lack sufficient evidence to say
it has changed. Bob’s weight could still be 190. The above sample mean
could have occurred by random chance with a probability of .0950 or 9.50%
of the time. What if 0t had been .10 instead of .05? We would then have sufficient
evidence to reject H0 and say it has changed. Example 4:
l. A shipment of machined parts has a critical dimension that is normally distributed with mean 12 centimeters and standard deviation 0.1
centimeters. The acceptance sampling team suspects that the dimension
is less than 12 centimeters. They take a simple random sample of 25 of
these parts and obtain a mean of 11.99. Is the acceptance sampling team
correct in their assertions? Use an or level of 0.01. {4 :ﬁm ‘2 h .2 2.5. SE .5? IL??? HA: a <2 :2
ﬂ law:2. m “30' was
nil .02.»
W5
PVq/UE ‘53 .3085 Ccmm'i rejﬁcl' Hm Confidence Intervals and TwoSided Tests:
A level 0t twosided significance test rejects a hypothesis H 0 : ,u = #0 exactly when the value :“0 falls outside a level 1—(1 confidence interval for ,u. Example using the weight on Bob’s drivers license: a,
Bob’s data: 193 194 192 191 Sample Mean = 192.5 == X It was assumed that 6 = 3 for the population of Bob’s weights.
Calculate 95% Confidence Interval: 192.5 + / — 1.960 (.3 / sqrt 4) Lecture 7, Chapter 6
Page 9 (189.56 , 195.44) From Page 8 and 9, a 2 side hypothesis test using 0. = .05 failed to reject Ho,
meaning that Bob’s weight could still be 190. You can see that this result is
consistent with the 95% confidence interval above, since 190 is included in
the confidence interval. If we repeated this example and calculated a 90% confidence interval we would get
(190.03, 194.97) If 0! = .10 the hypothesis would reject Ho meaning that Bob’s weight is
different from 190. You can see that this result is consistent with the 90%
confidence interval since 190 is not included in the interval. Two side hypotheses will reject H0 when a conﬁdence interval does not
include pro, provided that a and the conﬁdence level are equivalent. A 99% confidence level is equivalent to a =.01
A 95% conﬁdence level is equivalent to a: .05
A 90% confidence level is equivalent to a: .10 Section 6.3
Use and Abuse of Tests: Choosing a Level of Significance:
If we want to make a decision based on our test, we choose a level of significance in advance. We do not have to do this, however, if we are only interested in describing the strength of our evidence. If we do choose a level
of significance in advance, we must choose a by asking how much evidence
is required to reject H 0 . The choice of or depends on the type of study we are doing. If the value for a is not given, use a = .05 Some Cautions about Statistical tests: Lecture 7, Chapter 6
Page 10 “is As with CI’s, badly designed surveys or experiments often
produce invalid results. Formal statistical inference cannot
correct basic ﬂaws in data collection. As with CI’s, tests of significance are based on laws of
probability. Random sampling or random assignment of
subjects to treatments ensures that these laws apply.
Statistical significance is not the same thing as practical
significance. There is no sharp border between “significant” and “non
significant”, only increasingly strong evidence as the P—Value
gets smaller. It is possible that a nonsignificant result is due to the sample
size being too small. Larger sample sizes are capable of
detecting smaller shifts. Lecture 7, Chapter 6 Page 11 ...
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