population and sample mean
TRANSCRIPT
POPULATION AND SAMPLE MEAN Avjinder Singh Kaler and Kristi Mai
β’ Estimating a Population Mean
β’ π unknown
β’ π known
β’ Estimating the difference between two population means
β’ Independent samples
β’ Dependent samples
Main Ideas: β’ The sample mean is the best point estimate of the population mean
β’ We can use a sample mean to construct a C.I. to estimate the true value of a population mean
β’ We must learn how to find the sample size necessary to estimate a population mean
Recall:
β’ π₯ = π₯
π : sample mean
β’ π₯ targets π and is an individual value that is used as an estimate (i.e. it is a point estimate for π)
Notice: There are two situations when estimating a population mean 1. π, the population standard deviation, is known
2. π, the population standard deviation, is NOT known
β’ Margin of Error (estimating the population mean when π is known)
β’ πΈ = ππΌ/2 βπ
π
β’ Notice: The margin of error changes when what we are estimating changes!!
β’ Constructing a C.I.
β’ Requirements:
The sample must be a SRS
The value of π is known
The population is normal OR π > 30
β’ C.I.:
π₯ β πΈ < π < π₯ + πΈ
Same as: π₯ Β± πΈ and (π₯ β πΈ, π₯ + πΈ)
β’ Minimum required sample size
β’ Sample size needed: β’ If π is known:
π =(ππΌ/2) β π
πΈ
2
β’ If not a whole number, ALWAYS round up to the nearest whole number for minimum required sample sizes
Some Key Points: β’ The sample mean is still the best point estimate of the population
mean
β’ We can use a sample mean to construct a C.I. to estimate the true value of a population mean even when we do not know the population standard deviation
β’ We see that if requirements are generally met but π is unknown, we must use a t-distribution
The Student t Distribution:
β’ If a population has a normal distribution, then the following formula describes the t-distribution:
π‘ =π₯ βπ
π
π
β’ The above formula is a t-score; a measure of relative standing
β’ We are estimating the unknown population standard deviation with the sample standard deviation
β’ This estimation would typically lead to unreliability and so we compensate for this inherent unreliability with wider intervals and βfatter tailsβ displayed in the density curve
β’ We must utilize a t-table or t-calculator when using the t-distribution
β’ We NEED degrees of freedom
Degrees of Freedom (ππ) for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed upon all the data values
β’ Recall:
β’ π = π₯βπ₯ 2
πβ1 : sample standard deviation
β’ Margin of Error
πΈ = π‘πΌ/2 βπ
π with ππ = π β 1
Notice: The margin of error also changes when the information we have changes
Constructing a C.I. β’ Requirements:
β’ The sample must be a SRS
β’ The value of π is NOT known
β’ The population is normal OR π > 30
β’ C.I.: β’ π₯ β πΈ < π < π₯ + πΈ
β’ Same As: π₯ Β± πΈ and π₯ β πΈ, π₯ + πΈ
β’ Notice that the C.I. appears to be the same β however, it will NOT be the same as the previous CI for π because (with our uncertainty about π) the margin of error changed
β’ The student t distribution is different for different sample sizes
β’ The t distribution has the same general symmetric bell shape as the Normal distribution, but reflects the greater variability that is expected when samples are smaller
β’ The t distribution has a mean of π‘ = 0 just as the standard normal distribution has a mean of π§ = 0
Is the population normal OR is n>30
Is π known or unknown?
Use normal distribution
(Normal -Calculator)
Use t distribution
(t-calculator) Use nonparametric
method or bootstrapping
technique
Yes
No
Known
Unknown
Requirements: The sample must be a SRS The value of π is known The population is normal OR π > 30
Test Statistic: z =π₯ βπ
π
π
π: population mean (assumed true under π»0)
Note: p-values and critical values are from Z-table
Requirements:
The sample must be a SRS The value of π is NOT known
The population is normal OR π > 30
Test Statistic: t =π₯ βπ
π
π
; ππ = π β 1
π: population mean (assumed true
under π»0)
Note: p-values and critical values are from t-table
π known π NOT known
Listed below are the measured radiation emissions (in W/kg) corresponding to a sample of cell phones. Use a 0.05 level of significance to test the claim that cell phones have a mean radiation level that is less than 1.00 W/kg. The summary statistics are: .
0.38 0.55 1.54 1.55 0.50 0.60 0.92 0.96 1.00 0.86 1.46
0.938 and 0.423x s
Requirement Check: 1. We assume the sample is a simple random sample. 2. The sample size is n = 11, which is not greater than 30, so we must check
a normal quantile plot for normality.
Note: (See plot on the right) The points are reasonably close to a straight line and there is no other patter, so we conclude that The data appear to be from a normally distributed Population.
Step 1: The claim that cell phones have a mean radiation level less than 1.00 W/kg is expressed as ΞΌ < 1.00 W/kg.
Step 2: The alternative to the original claim is ΞΌ β₯ 1.00 W/kg.
Step 3: The hypotheses are written as:
Step 4: The stated level of significance is πΌ = 0.05.
Step 5: Because the claim is about a population mean ΞΌ, the statistic most relevant to this test is the sample mean:
0
1
: 1.00 W/kg
: 1.00 W/kg
H
H
x
Step 6: Calculate the test statistic and then find the P-value or the critical value using StatCrunch.
0.938 1.000.486
0.423
11
xxt
s
n
Step 7: Critical Value Method: Because the test statistic of t = β0.486 does not fall in the critical region bounded by the critical value of t = β1.812, fail to reject the null hypothesis.
Step 7: P-value method:
Using StatCrunch, the P-value computed is 0.3187. Since the P-value is greater than Ξ± = 0.05, we fail to reject the null hypothesis.
Step 8:
Because we fail to reject the null hypothesis, we conclude that there is not sufficient evidence to support the claim that cell phones have a mean radiation level that is less than 1.00 W/kg.
We can use a confidence interval for testing a claim about ΞΌ.
For a two-tailed test with a 0.05 significance level, we construct a 95% confidence interval.
For a one-tailed test with a 0.05 significance level, we construct a 90% confidence interval.
Using the cell phone example, construct a confidence interval that can be used to test the claim that ΞΌ < 1.00 W/kg, assuming a 0.05 significance level.
Note that a left-tailed hypothesis test with Ξ± = 0.05 corresponds to a 90% confidence interval.
Using StatCrunch, the confidence interval is:
0.707 W/kg < ΞΌ < 1.169 W/kg
Because the value of ΞΌ = 1.00 W/kg is contained in the interval, we fail to reject the null hypothesis that ΞΌ = 1.00 W/kg .
Based on the sample of 11 values, we do not have sufficient evidence to support the claim that the mean radiation level is less than 1.00 W/kg.
When Ο is known, we use test that involves the standard normal distribution.
In reality, it is very rare to test a claim about an unknown population mean
while the population standard deviation is somehow known.
The procedure is essentially the same as a t test, with the following
exception: The test statistic is
The P-value and critical values can be computed using StatCrunch.
xxz
n
If we repeat the cell phone radiation example, with the assumption that Ο = 0.480 W/kg, the test statistic is:
The example refers to a left-tailed test, so the P-value is the area to the left of z = β0.43, which is 0.3342.
Since the P-value is greater than πΌ = 0.05, we fail to reject the null and reach the same conclusion as before.
0.938 1.000.43
0.480
11
xxz
n
Main Ideas: β’ The sample mean is the best point estimate of the population mean
β’ We can use two independent sample means to construct a
confidence interval that can be used to estimate the true value of the
underlying difference in the corresponding population means
β’ We can also test claims about the difference between two population
means
Notation:
Dependent samples two samples are dependent if the
sample values are paired
Independent samples two samples are independent if
the sample values from one are
not related to or somehow
naturally paired/matched with the
sample values from the other
Requirements: β’ Population standard deviations (π1 and π2) are NOT known and
NOT assumed equal
β’ The two samples are independent
β’ Both samples are SRS
β’ Both π1 > 30 and π2 > 30 OR both samples come from populations that are normal
β’ Margin of Error
πΈ = π‘πΌ/2 βπ 12
π1+
π 22
π2 and ππ = min π1 β 1, π2 β 1
β’ C.I.: π₯ 1 β π₯ 2 β πΈ < π1 β π2 < π₯ 1 β π₯ 2 + πΈ
β’ Notice that we are often interested in whether or not 0 is included within the limits of the confidence interval constructed, i.e., whether or not π1 β π2 = 0 is reasonable
β’ Requirements:
β’ Requirements and degrees of freedom (df) are the same as in the C.I. before
β’ Test Statistic: π‘ =π₯ 1βπ₯ 2 β π1βπ2
π 12
π1+
π 22
π2
Researchers conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were given by a panel of judges. Researchers make the claim that βblue enhances performance on a creative taskβ. Test the claim using a 0.01 significance level.
Requirement check:
1. The values of the two population standard deviations are unknown and assumed not equal.
2. The subject groups are independent.
3. The samples are simple random samples.
4. Both sample sizes exceed 30.
The requirements are all satisfied.
The data:
Background color Sample size Sample mean Sample standard deviation
Red Background n = 35 s = 0.97
Blue Background n = 36 s = 0.63
3.39x
3.97x
Step 1: The claim that βblue enhances performance on a creative taskβ
can be restated as βpeople with a blue background (group 2) have a
higher mean creativity score than those in the group with a red background
(group 1)β. This can be expressed as ΞΌ1 < ΞΌ2.
Step 2: If the original claim is false, then ΞΌ1 β₯ ΞΌ2.
Step 3: The hypotheses can be written as:
OR π»0: π1βπ2=0π»1: π1βπ2<0
0 1 2
1 1 2
:
:
H
H
Step 4: The significance level is Ξ± = 0.05.
Step 5: Because we have two independent samples and we are testing a claim
about two population means, we use a t-distribution.
Step 6: Calculate the test statistic. 1 2 1 2
2 2
1 2
1 2
2 2
( ) ( )
(3.39 3.97) 02.979
0.97 0.63
35 36
x xt
s s
n n
Step 6: Because we are using a t-distribution, the critical value of t = β2.441 is found using StatCrunch. We use 34 degrees of freedom.
Step 7: Because the test statistic does fall in the critical region, we reject the
null hypothesis ΞΌ1 β ΞΌ2.
P-Value Method: StatCrunch provides a P-value, and the area to the left of
the test statistic of t = β2.979 is 0.0021. Since this is less than the significance
level of 0.01, we reject the null hypothesis.
Conclusion: There is sufficient evidence to support the claim that the red
background group has a lower mean creativity score than the blue
background group.
Using the data from this color creativity example, construct a 98% confidence interval estimate for the difference between the mean creativity score for those with a red background and the mean creativity score for those with a blue background.
Using StatCrunch, the 98% confidence interval obtained is:
β1.05 < π1 β π2 < β0.11
2 2 2 2
1 2/2
1 2
0.97 0.632.441 0.475261
35 36
s sE t
n n
1 23.39 and 3.97x x
1 2 1 2 1 2
1 2
( ) ( ) ( )
1.06 ( ) 0.10
x x E x x E
We are 98% confident that the limits β1.05 and β0.11 actually do contain the difference between the two population means.
Because those limits do not include 0, our interval suggests that there is a significant difference between the two means.
These methods are rarely used in practice because the underlying assumptions are usually not met.
1. The two population standard deviations are both known
β’ the test statistic will be a z instead of a t and use the standard normal model.
2. The two population standard deviations are unknown but assumed to be equal
β’ pool the sample variances
1 2 1 2
2 2
1 2
1 2
( ) ( )x xz
n n
The test statistic will be:
P-values and critical values are found using StatCrunch.
1 2 1 2 1 2( ) ( ) ( )x x E x x E
2 2
1 2/ 2
1 2
E zn n
The test statistic will be
Where the pooled sample variance is
with
1 2 1 2
2 2
1 2
( ) ( )
p p
x xt
s s
n n
2 22 1 1 2 2
1 2
( 1) ( 1)
( 1) ( 1)p
n s n ss
n n
1 2df 2n n
1 2 1 2 1 2( ) ( ) ( )x x E x x E
2 2
/2
1 2
p ps sE t
n n
1 2df 2n n
Independent Samples (Two Additional Methods)
β’ π1 πππ π2 known β Z Test / Z Interval
β’ π1 = π2 -- Pooled Sample Variance
Dependent Samples
β’ When samples are paired, we use a different methodology
Main Ideas: β’ The sample mean is still the best point estimate of the population mean
β’ We can use two dependent sample means to construct a confidence interval
that can be used to estimate the true value of the underlying difference in the
corresponding population means
β’ We can also test claims about the difference between two population means
β’ In experimental design, using dependent samples is generally better and more
practical than assuming two independent samples
Notation:
β’ π: π‘βπ πππππ£πππ’ππ ππππππππππ πππ‘π€πππ π‘π€π π£πππ’ππ ππ π π πππππ πππ‘πβππ ππππ
β’ π: π‘βπ ππ’ππππ ππ πππππ ππ πππ‘π
β’ ππ: π‘βπ ππππ’πππ‘πππ ππππ ππ π‘βπ πππππππππππ πππ πππ π‘βπ πππππ ππ πππ‘π
β’ π : π‘βπ π πππππ ππππ ππ π‘βπ πππππππππππ πππ π‘βπ ππππππ π πππππ πππ‘π
β’ π π: π‘βπ π πππππ π π‘ππππππ πππ£πππ‘πππ ππ π‘βπ πππππππππππ πππ π‘βπ ππππππ π πππππ πππ‘π
Requirements
β’ The sample data are dependent
β’ Both samples are SRS
β’ Either π > 30 OR the paired differences come from a population that is normal
Margin of Error
β’ πΈ = π‘πΌ/2 βπ π
π with ππ = π β 1
C.I.
β’ π β πΈ < ππ < π + πΈ
Notice that we are often interested in whether or not 0 is included within the limits of the confidence interval constructed, i.e., whether or not ππ = 0 is reasonable
Requirements:
β’ Requirements and Degrees of freedom (ππ) are the same as in the C.I. above
Test Statistic:
π‘ =π βππ
π ππ
Use the sample data below with a significance level of 0.05 to test the
claim that for the population of heights of presidents and their main
opponents, the differences have a mean greater than 0 cm (so presidents
tend to be taller than their opponents).
Height (cm) of President 189 173 183 180 179
Height (cm) of Main Opponent 170 185 175 180 178
Difference d 19 -12 8 0 1
Requirement Check:
1. The samples are dependent because the values are paired.
2. The pairs of data are randomly selected.
3. The number of data points is 5, so normality should be checked (and it is
assumed the condition is met).
Step 1: The claim is that Β΅d > 0 cm.
Step 2: If the original claim is not true, we have Β΅d β€ 0 cm.
Step 3: The hypotheses can be written as:
0
0
: 0 cm
: 0 cm
d
d
H
H
Step 4: The significance level is Ξ± = 0.05.
Step 5: We use the Student t-distribution.
The summary statistics are: 3.2
11.4
d
s
Step 6: Determine the value of the test statistic:
with df = 5 β 1 = 4
3.2 00.628
11.4
5
d
d
dt
s
n
Step 6: Using StatCrunch, the P-value is 0.282.
Using the critical value method:
Step 7: Because the P-value exceeds 0.05, or because the test statistic does not fall in the critical region, we fail to reject the null hypothesis.
Conclusion: There is not sufficient evidence to support the claim that for the population of heights of presidents and their main opponent, the differences have a mean greater than 0 cm.
In other words, presidents do not appear to be taller than their opponents.
Confidence Interval: Support the conclusions with a 90% confidence
interval estimate for Β΅d.
/2
11.42.132 10.8694
5
dsE t
n
3.2 10.8694 3.2 10.8694
7.7 14.1
d
d
d
d E d E
We have 90% confidence that the limits of β7.7 cm and 14.1 cm contain the true value of the difference in height (presidentβs height β opponentβs height).
See that the interval does contain the value of 0 cm, so it is very possible that the mean of the differences is equal to 0 cm, indicating that there is no significant difference between the heights.
Complete the following:
β’ Practice Problems 5
β’ Practice Problems 6