Download μ For given set of data, sample mean = 4.31 Thus, point estimate of

Test 3 Review Math 1342
1) A point estimate of the population mean (µ) is a sample mean.
For given set of data, x  sample mean = 67.7
Thus, point estimate of population mean ( ) is 67.7
2) A point estimate of the population mean (µ) is a sample mean.
For given set of data, x  sample mean = 4.31
Thus, point estimate of the population mean ( ) is 4.31
3) Confidence interval is an interval estimate of the population mean (µ)
4) Confidence interval is an interval estimate of the population mean (µ)
Level of Confidence = 99.74%
α = 100% - (Level of Confidence) = 0.26% = 0.0026
α/2 = 0.13% = 0.0013
Calculate zα/2 by using standard normal distribution with α/2 = 0.13% = 0.0013 as right-tailed
area and
mean = 0 and standard deviation = 1:
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 2:
a) Form all possible samples o si e 1 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
) 99.74% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (4.61175, 7.62325)
ontains the population mean (μ) or not, we are only 99.74% confident that (4.61175, 7.62325)
ontains the population mean (μ).
5) Confidence interval is an interval estimate of the population mean (µ)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 1 :
a) Form all possible samples o si e 32 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
) 95.44% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (74.2519413349199, 85.5605586650801)
ontains the population mean (μ) or not, we are only 95.44% on ident that
(74.2519413349199, 85.5605586650801)
ontains the population mean (μ).
6) Confidence interval is an interval estimate of the population mean (µ)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 1 :
a) Form all possible samples o si e 2 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
c) 95.44% of these confidence intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (37.3827653396796, 82.6172346603204)
ontains the population mean (μ) or not, we are only 95.44% on ident that
(37.3827653396796, 82.6172346603204)
contains the population mean (μ).
7) Confidence interval is an interval estimate of the population mean (µ)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 0.7:
a) Form all possible samples of size 15 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
) 95.44% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (3.79201755298752, 4.51464911367848)
ontains the population mean (μ) or not, we are only 95.44% on ident that
(3.79201755298752, 4.51464911367848)
ontains the population mean (μ).
8)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 29:
a) Form all possible samples o si e 10 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
) 90% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (521.366762466525, 530.633237533475)
ontains the population mean (μ) or not, we are only 90% confident that (521.366762466525,
530.633237533475)
ontains the population mean (μ).
9)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and standard deviation (σ) = 0.28:
a) Form all possible samples o si e 54 and al ulate sample mean ( ) of each sample.
b) For each sample mean, calculate a confidence interval; hence, there are many, many
confidence intervals.
) 95% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (1.29531777939781, 1.44468222060219)
ontains the population mean (μ) or not, we are only 95% on ident that (1.29531777939781,
1.44468222060219)
ontains the population mean (μ).
10)
Level of confidence = 90%
α = 100% - (Level of confidence) = 10% = 0.10
E = Margin of Error = z /2
E = Margin of Error = z /2

n

 2.5 
 1.64 
  0.693
n
 35 
Interpretation:
The margin of error controls the width of the confidence interval.
In constructing confidence, we have control over the level of confidence;
hence, control over α. We also have ontrol over the sample size (n).
Note that high confidence level means low α.
And Low α means higher
| z / 2 |
The larger the margin of error, the wider the confidence interval will be;
however, wider confidence interval means less precision in estimating
the population mean .
The smaller the margin of error, the narrower the confidence interval will be;
however, narrower confidence interval means more precision in estimating
the population mean .
11)
E = Margin of Error = z /2
E = Margin of Error = z /2


n
 870 
 2.58 
  260.93
n
 74 
Interpretation:
The margin of error controls the width of the confidence interval.
In constructing confidence, we have control over the level of confidence;
hence, control over α. We also have ontrol over the sample size (n).
Note that high confidence level means low α.
And Low α means higher
| z / 2 |
The larger the margin of error, the wider the confidence interval will be;
however, wider confidence interval means less precision in estimating
the population mean .
The smaller the margin of error, the narrower the confidence interval will be;
however, narrower confidence interval means more precision in estimating
the population mean .
12) E = z /2

n
 z  
n =   /2 
 E 
2
13) E = z /2

n
 z  
n =   /2 
 E 
2
Using computational tool on www.simulation-math.com
Note: 1%/2 = 0.5% = 0.005
Middle Area = 95%
5%/2 = 2.5% = 0.025
17)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and unknown population standard deviation:
a) Form all possible samples o si e 12 and al ulate sample mean ( ) and sample standard
deviation (s) of each sample.
b) For each pair of sample mean and standard deviation, calculate a confidence interval; hence,
there are many, many confidence intervals.
) 95% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (232.707409319578, 253.292590680422)
ontains the population mean (μ) or not, we are only 95% on ident that (232.707409319578,
253.292590680422)
ontains the population mean (μ).
18)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and unknown population standard deviation:
a) Form all possible samples o si e 30 and al ulate sample mean ( ) and sample standard
deviation (s) of each sample.
b) For each pair of sample mean and standard deviation, calculate a confidence interval; hence,
there are many, many confidence intervals.
) 99% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (75.90472530892, 90.09527469108)
contains the population mean (μ) or not, we are only 99% on ident that (75.90472530892,
90.09527469108)
ontains the population mean (μ).
19)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and unknown population standard deviation:
a) Form all possible samples o si e 2 and al ulate sample mean ( ) and sample standard
deviation (s) of each sample.
b) For each pair of sample mean and standard deviation, calculate a confidence interval; hence,
there are many, many confidence intervals.
) 95% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (67.5565147412143, 84.8434852587857)
contains the population mean (μ) or not, we are only 95% on ident that ( 7.55 5147412143,
84.8434852587857)
ontains the population mean (μ).
20)
Interpretation of a confidence interval:
For a population with unknown mean (μ) and unknown population standard deviation:
a) Form all possible samples o si e 14 and al ulate sample mean ( ) and sample standard
deviation (s) of each sample.
b) For each pair of sample mean and standard deviation, calculate a confidence interval; hence,
there are many, many confidence intervals.
) 90% o these on iden e intervals will ontain the population mean (μ).
d) Since we do not know if the confidence interval (523.434743858437, 804.845256141563)
contains the population mean (μ) or not, we are only 90% on ident that (523.434743858437,
804.845256141563)
ontains the population mean (μ).
21)
22) Find sample size (n) given margin of error (E)
23) Hypothesis Testing Logic:
 Some claim is made about the population (  ) mean
 Due to various reasons, we think the population mean
different than what is claimed.
 Take a sample of size n and calculate sample mean
then compare sample mean
 Based on difference of
(  ) is
(x );
( x ) with population mean (  ) .
x and  , we will decide whether or not
to accept or reject what is claimed about the population mean.
 Difference between x and  is significant or not depends on the
level of significance of the hypothesis test, which is specified by
the person performing the hypothesis testing.
 = level of significance of the hypothesis test 
 If large difference between

x and  can be accepted, then
should be small (around 0.01 = 1%).
 If difference between x and  must be small, then

should be large (around 0.10 = 10%).
 In traditional hypothesis testing, decision about accepting or rejecting
what is claimed about the population mean is based on the following:
Test Statistic =
x 




n

Calculated based on  and normal distribution if  is known

Critical Value = 
Calculated based on  and t-distribution if  is unknown

24)
25)
t-score can also be found by using computational tool on www.simulation-math.com
Thus, we cannot reject the null hypothesis Ho: µ = 0.4
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