beljuA

2021-10-20

The population mean may be reliably estimated using a statistic calculated from data recorded from randomly selected individuals.’ Explain this statement with reference to sampling distributions and the central limit theorem.

Alara Mccarthy

Skilled2021-10-21Added 85 answers

Step 1

Given information:

The population mean may be reliably estimated using a statistic calculated from data recorded from selected individuals.

Step 2

Sampling distribution of sample mean:

Let a particular characteristic of a population is of interest in a study. Denote µ as the population mean of that characteristic and σ as the population standard deviation of that characteristic.

Now, it is not always possible to study every population unit. So, a sample of size n is taken from the population.

Let X denotes the random variable that measures the particular characteristic of interest. Let,$X}_{1},{X}_{2},\dots ,{X}_{n$ be the values of the random variable for the n units of the sample from a normal distribution.

Then, the sample mean has a sampling distribution that has population mean or expected value same as that of X and population standard deviation, called the standard error, which is the population standard deviation of X divided by the square root of the sample size n. The parameters are as follows:

$E\left(\stackrel{\u2015}{X}\right)=E(\frac{1}{n}\sum {X}_{i})$

$=\mu$

$SD\left(\stackrel{\u2015}{X}\right)=\frac{\sigma}{\sqrt{n}}$ .

Step 3

Central limit theorem:

Central limit theorem states that, the shape of the sampling distribution of sample mean will be approximately normal, if the sample size is large and the standard deviation of the population is finite.

If$X}_{1},{X}_{2},\dots ,{X}_{n$ are independent and identically distributed (iid) random variables with finite mean, $E\left({X}_{i}\right)=\mu$ and finite variance $V\left({X}_{i}\right)={\sigma}^{2}$ , then Z will converge to the standard normal distribution when the sample size, n tends to infinity, where:

$Z=\frac{\stackrel{\u2015}{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)$

As a rule of thumb if the selected sample size is greater than 30, then the sample mean will be approximately normally distributed.

Thus, the sample mean will be unbiased estimator of the population mean.

Given information:

The population mean may be reliably estimated using a statistic calculated from data recorded from selected individuals.

Step 2

Sampling distribution of sample mean:

Let a particular characteristic of a population is of interest in a study. Denote µ as the population mean of that characteristic and σ as the population standard deviation of that characteristic.

Now, it is not always possible to study every population unit. So, a sample of size n is taken from the population.

Let X denotes the random variable that measures the particular characteristic of interest. Let,

Then, the sample mean has a sampling distribution that has population mean or expected value same as that of X and population standard deviation, called the standard error, which is the population standard deviation of X divided by the square root of the sample size n. The parameters are as follows:

Step 3

Central limit theorem:

Central limit theorem states that, the shape of the sampling distribution of sample mean will be approximately normal, if the sample size is large and the standard deviation of the population is finite.

If

As a rule of thumb if the selected sample size is greater than 30, then the sample mean will be approximately normally distributed.

Thus, the sample mean will be unbiased estimator of the population mean.

The product of the ages, in years, of three (3) teenagers os 4590. None of the have the sane age. What are the ages of the teenagers???

Use the row of numbers shown below to generate 12 random numbers between 01 and 99

78038 18022 84755 23146 12720 70910 49732 79606

Starting at the beginning of the row, what are the first 12 numbers between 01 and 99 in the sample?How many different 10 letter words (real or imaginary) can be formed from the following letters

H,T,G,B,X,X,T,L,N,J.Is every straight line the graph of a function?

For the 1s orbital of the Hydrogen atom, the radial wave function is given as: $R(r)=\frac{1}{\sqrt{\pi}}(\frac{1}{{a}_{O}}{)}^{\frac{3}{2}}{e}^{\frac{-r}{{a}_{O}}}$ (Where ${a}_{O}=0.529$ ∘A)

The ratio of radial probability density of finding an electron at $r={a}_{O}$ to the radial probability density of finding an electron at the nucleus is given as ($x.{e}^{-y}$). Calculate the value of (x+y).Find the sets $A$ and $B$ if $\frac{A}{B}=\left(1,5,7,8\right),\frac{B}{A}=\left(2,10\right)$ and $A\cap B=\left(3,6,9\right)$. Are they unique?

What are the characteristics of a good hypothesis?

If x is 60% of y, find $\frac{x}{y-x}$.

A)$\frac{1}{2}$

B)$\frac{3}{2}$

C)$\frac{7}{2}$

D)$\frac{5}{2}$The numbers of significant figures in $9.1\times {10}^{-31}kg$ are:

A)Two

B)Three

C)Ten

D)Thirty oneWhat is positive acceleration?

Is power scalar or vector?

What is the five-step process for hypothesis testing?

How to calculate Type 1 error and Type 2 error probabilities?

How long will it take to drive 450 km if you are driving at a speed of 50 km per hour?

1) 9 Hours

2) 3.5 Hours

3) 6 Hours

4) 12.5 HoursWhat is the square root of 106?