What is the mathematical formula for the pooled variance of two populations?

Jefferson Booth
2022-11-18
Answered

What is the mathematical formula for the pooled variance of two populations?

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Kaeden Lara

Answered 2022-11-19
Author has **23** answers

The pooled standard deviation is given by

$S}_{p}=\sqrt{\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}$

for variance remove the sqare root operator

$V}_{p}=\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2$

$S}_{p}=\sqrt{\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}$

for variance remove the sqare root operator

$V}_{p}=\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2$

asked 2022-09-28

The denominator of the t score formula for the independent-samples t-test is ______.

1 - population standard error

2 - estimated pooled standard deviation

3 - estimated pooled variance

4 - estimated standard error

1 - population standard error

2 - estimated pooled standard deviation

3 - estimated pooled variance

4 - estimated standard error

asked 2022-08-11

Consider the data with analysis shown in the following computer output:

$\begin{array}{|cccc|}\hline \text{Level}& \text{N}& \text{Mean}& \text{StDev}\\ \text{A}& 4& 10.500& 2.902\\ \text{B}& 5& 16.800& 2.168\\ \text{C}& 6& 10.800& 2.387\\ \hline\end{array}$

$\begin{array}{|cccccc|}\hline \text{Source}& \text{DF}& \text{SS}& \text{MS}& \text{F}& \text{P}\\ \text{Groups}& 2& 125.07& 62.54& 10.34& 0.002\\ \text{Error}& 12& 72.55& 6.05\\ \text{Total}& 14& 197.62\\ \hline\end{array}$

What is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means and differences in means?

Round your answer for the pooled standard deviation to two decimal places.

The pooled standard deviation is =

degrees of freedom =

$\begin{array}{|cccc|}\hline \text{Level}& \text{N}& \text{Mean}& \text{StDev}\\ \text{A}& 4& 10.500& 2.902\\ \text{B}& 5& 16.800& 2.168\\ \text{C}& 6& 10.800& 2.387\\ \hline\end{array}$

$\begin{array}{|cccccc|}\hline \text{Source}& \text{DF}& \text{SS}& \text{MS}& \text{F}& \text{P}\\ \text{Groups}& 2& 125.07& 62.54& 10.34& 0.002\\ \text{Error}& 12& 72.55& 6.05\\ \text{Total}& 14& 197.62\\ \hline\end{array}$

What is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means and differences in means?

Round your answer for the pooled standard deviation to two decimal places.

The pooled standard deviation is =

degrees of freedom =

asked 2022-05-09

Samples are taken from two different types of honey and the viscosity is measured.

Honey A:

Mean: 114.44

S.D : 0.62

Sample Size: 4

Honey B:

Mean: 114.93

S.D: 0.94

Sample Size: 6

Assuming normal distribution, test at 5% significance level whether there is a difference in the viscosity of the two types of honey?

Here's what I did:

I took my null hypothesis as $\mu $B - $\mu $A = 0 and alternative hypothesis as $\mu $B - $\mu $A $\ne $ 0

Then I did my calculations which were as following:

Test Statistic = (B -A ) - ($\mu $B - $\mu $A) / sqrt {(variance B / sample size B) + (variance A / sample size A)}

This gave me test statistic as = 0.49/0.49332 that is equal to 0.993

However the test statistic in the book solution is given as 0.91. What am I doing wrong?

Honey A:

Mean: 114.44

S.D : 0.62

Sample Size: 4

Honey B:

Mean: 114.93

S.D: 0.94

Sample Size: 6

Assuming normal distribution, test at 5% significance level whether there is a difference in the viscosity of the two types of honey?

Here's what I did:

I took my null hypothesis as $\mu $B - $\mu $A = 0 and alternative hypothesis as $\mu $B - $\mu $A $\ne $ 0

Then I did my calculations which were as following:

Test Statistic = (B -A ) - ($\mu $B - $\mu $A) / sqrt {(variance B / sample size B) + (variance A / sample size A)}

This gave me test statistic as = 0.49/0.49332 that is equal to 0.993

However the test statistic in the book solution is given as 0.91. What am I doing wrong?

asked 2022-06-17

Assume that n= 20 and P =0.80 what is the standard deviation

asked 2022-07-22

Comparison between two diets in obese children, one low-fat and one low-carb 8 week intervention with the outcome of weight loss. Based on a similar study in adults, 20% of the children will not complete the study. For a 95% confidence interval with a margin of error of no more than 3 lbs., how many children should be recruited? In the adult trial, the low-fat and low-carb groups had a standard deviations of 8.4 and 7.7, respectively and each group had 100 participants.

A. Calculate the pooled standard deviation (Sp).

B. Calculate the number of children will need to be recruited for the trial if all finish the study use the Sp calculated for the standard deviation?

C. How many children would be needed to account for 20% attrition.

A. Calculate the pooled standard deviation (Sp).

B. Calculate the number of children will need to be recruited for the trial if all finish the study use the Sp calculated for the standard deviation?

C. How many children would be needed to account for 20% attrition.

asked 2022-06-05

Suppose we're studying the time it takes for a certain industrial process to complete. A recent study, which measured the time it took to complete 51 processes, gave a mean time of 2.396 minutes with standard deviation of 1.967 minutes. From past studies it has been observed that the standard deviation of the time it takes for the process to complete is 2.1 minutes. Calculate a confidence interval of 98% for the standard deviation of the time it takes to complete the process.

I don't understand why would they give me the value of the standard deviation of past processes, if I already have a value for ${s}^{2}$ (namely 1.967). By the way, I build the confidence interval by using the fact that $(n-1){S}^{2}/{\sigma}^{2}$ follows a Chi squared distribution and then calculating the probability of ${\sigma}^{2}$ being between the inferior and superior limits of the ${\chi}^{2}$ distribution for which the area under the curve is equal to (1−0.98)/2. I do not know why is that past value there if the interval can be calculated using only the first value.

I remember that in class, the professor said that ${S}_{p}^{2}$ is the single best estimator of the variance of a single population in terms of the variance of another population, though I didn't understand that. I suspect I can use the pooled variance in order to calculate a more accurate confidence interval taking in account both values of the standard deviation that they give me?

And in general, what sense does it make to calculate the confidence interval for a variable whose value I already know?

I don't understand why would they give me the value of the standard deviation of past processes, if I already have a value for ${s}^{2}$ (namely 1.967). By the way, I build the confidence interval by using the fact that $(n-1){S}^{2}/{\sigma}^{2}$ follows a Chi squared distribution and then calculating the probability of ${\sigma}^{2}$ being between the inferior and superior limits of the ${\chi}^{2}$ distribution for which the area under the curve is equal to (1−0.98)/2. I do not know why is that past value there if the interval can be calculated using only the first value.

I remember that in class, the professor said that ${S}_{p}^{2}$ is the single best estimator of the variance of a single population in terms of the variance of another population, though I didn't understand that. I suspect I can use the pooled variance in order to calculate a more accurate confidence interval taking in account both values of the standard deviation that they give me?

And in general, what sense does it make to calculate the confidence interval for a variable whose value I already know?

asked 2022-09-27

Find the pooled estimate of the standard deviation if n1 = 18, n2 = 14, S1= 3.3, and S2= 2.5.

(Write two decimal places)

(Write two decimal places)