Question

3729 Assignment 1 1/23/2019 DUE FEB 4th The mean and standard deviation of a sample are 68.5 and 11.2. Find the 95% confidence interval for this group? What is the assumption you made in this calculation? ● 2. State central limit theorem. How is it used in constructing the Z test?

Answer 1

1. Assumption: Bell shaped data

[ Applying Empirical Rule of Standard Deviation ]

95% Confidence Interval = $[-2\sigma + \bar{x} , \bar{x} + 2\sigma]$

= $[ -2*11.2 + 68.5 , 58.5 + 2*11.2 ]$

= $[ 46.1 , 90.9]$

2. Central Limit Theorem:

For large sample size(n > 30), the distribution is assumed to be normal,

the sample mean $\mu_{\bar{x}}$ = the population mean $\mu$

and the standard deviation of sample mean $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

where $\sigma$ = standard deviation of population.

$z = \frac{x-\mu}{\sigma}$

where z is the z-score, $x$ is the value of the element, μ is the population mean, and σ is the standard deviation.

Z - test using Central Limit Theorem : It changes population with mean μ and standard deviation σ into sample mean $\mu_{\bar{x}}$ and sample  standard deviation  $\sigma_{\bar{x}}$.

Here, $z = \frac{x - {\mu_{\bar{x}}}} {\sigma_{\bar{x}}}$