In a problem of comparing means of two populations we may come across the following conditions depending on the property of data.
(a)Pooled variance: when the data selected for comparing two population variance.
Then we used polled variance to draw interefences and compare the means.
It \(s_{1}^{2}\) and \(s_{2}^{2}\) are the sample variances of two population with equal population variances, then "pooled variance" is given by
\(\displaystyle{{s}_{{{p}}}^{{{2}}}}\ =\ {\frac{{{\left({n}_{{{1}}}\ -\ {1}\right)}\ {{s}_{{{1}}}^{{{2}}}}\ +\ {\left({n}_{{{2}}}\ -\ {1}\right)}\ {{s}_{{{2}}}^{{{2}}}}}}{{{n}_{{{1}}}\ +\ {n}_{{{2}}}\ -\ {2}}}}\)
In this case, the test statistic for comparing means is given by
\(\displaystyle{A}\ =\ {\frac{{\overline{{{x}_{{{1}}}}}-\overline{{{x}_{{{2}}}}}}}{{{s}_{{{p}}}\sqrt{{{\frac{{{1}}}{{{n}_{{1}}}}}+{\frac{{{1}}}{{{n}_{{2}}}}}}}}}}\ \text{with}\ {\left({n}_{{1}}\ +\ {n}_{{2}}\ -\ {2}\right)}\ {d}{f}\)
(b)Non - pooled variance:
On the contrary the property discused in part(a) about the data, when the populations variances of two populations are assumed to be unequal, then non-pooled variance is used to compare the means of two population.
It is given as
\(\displaystyle{S}\ \text{non-pooled}\ =\ \sqrt{{{\frac{{{{s}_{{1}}^{{2}}}}}{{{n}_{{1}}}}}\ +\ {\frac{{{{s}_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}\)
and the test statistic is
\(\displaystyle{A}\ =\ {\frac{{\overline{{{x}_{{{1}}}}}-\overline{{{x}_{{{2}}}}}}}{{{s}_{{{p}}}\sqrt{{{\frac{{{{s}_{{1}}^{{2}}}}}{{{n}_{{1}}}}}+{\frac{{{{s}_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}}}}`\)
(c)Paired interfence(Pained test):
Let us consider the case when
(i) sample a sizes are equal and
(ii) the samples are not independent but the sample observations are painted together.
i.e. the pair of observations\(\displaystyle{\left({x},{i}{y}.{i}\right)}\ {\left({i}={1},{2}..{n}\right)}\)
comprresponding to the same i-th unit.
For example. Suppose we want to test the effiency of a particular drug, say, for inducing sleep.Let xi and \(\displaystyle{y}{i}\ {\left({i}\ =\ {1}..{n}\right)}\) be the readings, in hours of sleep, an the i-th individual before and after the drug is givenrespectively.
Question
Provide three examples of studies when we can use (a) the pooled, (b) non-pooled, and (3) paired inference to compare means of two populations.
Answers (1)
2020-12-26
Relevant Questions
asked 2020-11-24
When should we use pooled, non-pooled or paired output to compare the means of two populations. you need to be specific and give the correct answer.
asked 2021-01-28
Indicate true or false for the following statements. If false, specify what change will make the statement true.
a) In the two-sample t test, the number of degrees of freedom for the test statistic increases as sample sizes increase.
b) When the means of two independent samples are used to to compare two population means, we are dealing with dependent (paired) samples.
c) The \(\displaystyle{x}^{{{2}}}\) distribution is used for making inferences about two population variances.
d) The standard normal (z) score may be used for inferences concerning population proportions.
e) The F distribution is symmetric and has a mean of 0.
f) The pooled variance estimate is used when comparing means of two populations using independent samples.
g) It is not necessary to have equal sample sizes for the paired t test.
a) In the two-sample t test, the number of degrees of freedom for the test statistic increases as sample sizes increase.
b) When the means of two independent samples are used to to compare two population means, we are dealing with dependent (paired) samples.
c) The \(\displaystyle{x}^{{{2}}}\) distribution is used for making inferences about two population variances.
d) The standard normal (z) score may be used for inferences concerning population proportions.
e) The F distribution is symmetric and has a mean of 0.
f) The pooled variance estimate is used when comparing means of two populations using independent samples.
g) It is not necessary to have equal sample sizes for the paired t test.
asked 2021-08-10
Suppose that you want to perform a hypothesis test based on independent random samples to compare the means of two populations. For each part, decide whether you would use the pooled t-test, the nonpooled t-test, the Mann– Whitney test, or none of these tests if preliminary data analyses of the samples suggest that the two distributions of the variable under consideration are
a. normal but do not have the same shape.
b. not normal but have the same shape.
c. not normal and do not have the same shape. both sample sizes are large.
asked 2020-12-25
True // False: comparing means. Determine if the following statements are true or false, and explain your reasoning for statements you identify as false.
(a) When comparing means of two samples where \(n_{1} = 20\)
and \(n_{2} = 40\),
we can use the normal model for the difference in means since \(n_{2} \geq 30.\)
(b) As the degrees of freedom increases, the t-distribution approaches normality.
(c) We use a pooled standard error for calculating the standard error of the difference between means when sample sizes of groups are equal to each other.
(a) When comparing means of two samples where \(n_{1} = 20\)
and \(n_{2} = 40\),
we can use the normal model for the difference in means since \(n_{2} \geq 30.\)
(b) As the degrees of freedom increases, the t-distribution approaches normality.
(c) We use a pooled standard error for calculating the standard error of the difference between means when sample sizes of groups are equal to each other.
