Question

- Suppose random samples of the same size are taken from two different populations with the same mean but different standard deviations. If you make two confidence intervals, one from each sample, and these intervals have the same confidence level, how will the confidence intervals differ? Explain.

Answer #1

To understand the problem. let us consider a situation,

Suppose we want to construct
confidence interval for population mean **µ of normal
population as,**

Width of confidence interval =

You can see that that the width of a
confidence interval is directly proportion to i.e the standard
deviation. **Thus, if we have two different populations with
the same mean but different standard deviations and it is need to
make two confidence intervals, one from each sample, and these
intervals have the same confidence level, Condidence interval will
be larger for such sample whose standard deviation is larger and
Condidence interval will be smaler for such sample whose standard
deviation is smaller.**

Two
samples were taken to see if the standard deviations two
populations are the same or not.
Sample1:
-size of sample is 21
-sample variance is 120
Sample 2:
-sizo of sample is 16
-Sample variance is 105
conduct the appropriate hypothesis test using a 10% level of
significance.

consider the following results frmo two independent random
samples taken from two populations. assume that the variances are
NOT equal.
Population 1
population 2
sample size
50
50
sample mean
35
30
sample variance
784
100
a) what is the "degrees of freedom" for these data?
b) what is the 95% confidence interval difference of the
population means?

Two different simple random samples are drawn from two different
populations. The first sample consists of
2020
people with
1111
having a common attribute. The second sample consists of
22002200
people with
15801580
of them having the same common attribute. Compare the results
from a hypothesis test of
p 1p1equals=p 2p2
(with a
0.050.05
significancelevel) and a
9595%
confidence interval estimate of
p 1p1minus−p 2p2.

Two different simple random samples are drawn from two different
populations. The first sample consists of 30 people with 14 having
a common attribute. The second sample consists of 1800 people with
1294 of them having the same common attribute. Compare the results
from a hypothesis test of p1= p2 (with a 0.05 significance level)
and a 95% confidence interval estimate of p1−p2.
Identify hypothesis, t statistic, critical value, p value

Two different simple random samples are drawn from two different
populations. The first sample consists of 30 people with 15 having
a common attribute. The second sample consists of 1900 people with
1379 of them having the same common attribute. Compare the results
from a hypothesis test of p1=p2 (with a 0.01 significance level)
and a 99% confidence interval estimate of p1−p2.
Find hypothesis, test statistic, critical value, p value, and
95% CL.

Two random samples are selected from two independent
populations. A summary of the samples sizes, sample means, and
sample standard deviations is given below:
n1=41, n2=44, x¯1=52.3, x¯2=77.3, s1=6 s2=10.8
Find a 96.5% confidence interval for the difference μ1−μ2 of the
means, assuming equal population variances.
Confidence Interval =

Two random samples are selected from two independent
populations. A summary of the samples sizes, sample means, and
sample standard deviations is given below:
n1=39,n2=40,x¯1=50.3,x¯2=73.8,s1=6s2=6.1
Find a 98% confidence interval for the difference μ1−μ2 of the
population means, assuming equal population variances.

Two random samples are selected from two independent
populations. A summary of the samples sizes, sample means, and
sample standard deviations is given below:
n1=51,n2=36,x¯1=56.5,x¯2=75.3,s1=5.3s2=10.7n1=51,x¯1=56.5,s1=5.3n2=36,x¯2=75.3,s2=10.7
Find a 97.5% confidence interval for the difference μ1−μ2μ1−μ2
of the means, assuming equal population variances.
Confidence Interval =

The following statistics were calculated from two random samples
taken from two populations following a normal distribution:
S1^2=350, n1=30, S2^2=700, n2=30
1. Can you infer that the two parent variances are different at
a 5% significance level?
2. If the number of samples is n1 = 15 and n2 = 15, repeat
question 1.
3. Describe what happens to the test statistic when the number
of samples decreases.

Consider the following data for two independent random samples
taken from two normal populations.
Sample 1
10
7
13
7
9
8
Sample 2
8
7
8
4
6
9
(a)Compute the two sample means.
Sample 1:
Sample 2:
(b)Compute the two sample standard deviations. (Round your
answers to two decimal places.)
Sample 1:
Sample 2:
(c) What is the point estimate of the difference between the two
population means? (Use Sample 1 − Sample 2.)
(d) What is the...

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