A random sample of 90 observations produced a mean of ?⎯⎯⎯=26.3 from a population with a normal distribution and a standard deviation ?=2.08. (a) Find a 90% confidence interval for u(b) Find a 99% confidence interval for u(c) Find a 95% confidence interval for u

The given data: Mean, x¯=26.3 Standard deviation,σ=2.08 Sample of observation, n=90 (a) 90%…

Answered: A random sample of 90 observations…

A First Course in Probability (10th Edition)

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Author:Sheldon Ross

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Question

A random sample of 90 observations produced a mean of ?⎯⎯⎯=26.3 from a population with a normal distribution and a standard deviation ?=2.08.

(a) Find a 90% confidence interval for u
(b) Find a 99% confidence interval for u
(c) Find a 95% confidence interval for u

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

The given data:

Mean, $\bar{x} = 26.3$

Standard deviation, $σ = 2.08$

Sample of observation, $n = 90$

(a) 90% confidence interval for $μ$ is given as:

$\begin{array}{rcl} 1 - α & = & 0.9 \\ α & = & 0.1 \end{array}$

z score of $\frac{α}{2}$ is given as:

$\begin{array}{rcl} z s c o r e (\frac{α}{2}) & = & z s c o r e (\frac{0.1}{2}) \\ = & 1.645 \end{array}$

The confidence interval is given as:

$C I = \bar{x} \pm z \frac{σ}{\sqrt{n}}$

Substitute the values into the above equation.

$\begin{array}{rcl} C I & = & \bar{x} \pm z \frac{σ}{\sqrt{n}} \\ = & 26.3 \pm 1.645 (\frac{2.08}{\sqrt{90}}) \\ = & 26.3 \pm 0.36067 \\ = & (25.94, 26.661,) \end{array}$

Therefore, the 90% confidence interval is (25.94, 26.661)

Step 2

(b) 99% confidence interval for $μ$ is given as:

$\begin{array}{rcl} 1 - α & = & 0.99 \\ α & = & 0.01 \end{array}$

z score of $\frac{α}{2}$ is given as:

$\begin{array}{rcl} z s c o r e (\frac{α}{2}) & = & z s c o r e (\frac{0.01}{2}) \\ = & 2.575 \end{array}$

The confidence interval is given as:

$C I = \bar{x} \pm \pm z \frac{σ}{\sqrt{n}}$

Substitute the values into the above equation.

$\begin{array}{rcl} C I & = & \bar{x} \pm z \frac{σ}{\sqrt{n}} \\ = & 26.3 \pm 2.575 (\frac{2.08}{\sqrt{90}}) \\ = & 26.3 \pm 0.56457 \\ = & (25.735, 26.865) \end{array}$

Therefore, the 99% confidence interval is ( 25.735, 26.865)

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