Normal Distribution Worksheet Find the highest and lowest interval that contains about 95% of the distribution. Assume the variables are continuous, random and normally distributed. m = 15, s = 3 m = 19, s = 2.5 m = 4.2, s = .25 m = 15, s = .085 Application of z-scores A brand of automobile tire has a life expectancy that is normally distributed, with a mean life of 30,000 miles and a standard deviation of 2,500 miles. The lifespan of three randomly selected tires are 33,000 miles, 24,750 miles, and 30,000 miles. Find the z-score that corresponds to each lifespan. Your company manufactures ball bearings. The diameters of the ball bearings are normally distributed, with a mean of 3 inches and a standard deviation of 0.02 inch. Find the z-scores of four randomly selected ball bearings with the diameters 3.01, 2.97, 2.98, and 3.05 inches. If you fill an order for 50,000 ball bearings and are willing to replace any whose diameter varies from 3 inches by more than 0.025 inch, how many ball bearings should you expect to replace? The lifespan of a species of fruit fly is normally distributed with a mean of 36 days and a standard deviation of 4 days. The lifespan of three randomly selected flies are 32 days, 35 days, 36 days, and 42 days. Find the z-score that corresponds to each lifespan.

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

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Normal Distribution Worksheet

Find the highest and lowest interval that contains about 95% of the distribution. Assume the variables are continuous, random and normally distributed.

m = 15, s = 3
m = 19, s = 2.5
m = 4.2, s = .25
m = 15, s = .085

Application of z-scores

A brand of automobile tire has a life expectancy that is normally distributed, with a mean life of 30,000 miles and a standard deviation of 2,500 miles. The lifespan of three randomly selected tires are 33,000 miles, 24,750 miles, and 30,000 miles. Find the z-score that corresponds to each lifespan.
Your company manufactures ball bearings. The diameters of the ball bearings are normally distributed, with a mean of 3 inches and a standard deviation of 0.02 inch.
1. Find the z-scores of four randomly selected ball bearings with the diameters 3.01, 2.97, 2.98, and 3.05 inches.
2. If you fill an order for 50,000 ball bearings and are willing to replace any whose diameter varies from 3 inches by more than 0.025 inch, how many ball bearings should you expect to replace?
The lifespan of a species of fruit fly is normally distributed with a mean of 36 days and a standard deviation of 4 days. The lifespan of three randomly selected flies are 32 days, 35 days, 36 days, and 42 days. Find the z-score that corresponds to each lifespan.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

The given values

1. $μ = 15, σ = 3$

When 95% level of significance the z value is 1.96.

Therefore, the 95% confidence interval is computed as follows:

$\begin{array}{rcl} C . I & = & μ \pm z σ \\ = & 15 \pm 1.96 \times 3 \\ = & (9.12, 20.88) \end{array}$

2. $μ = 19, σ = 2.5$

$\begin{array}{rcl} C . I & = & μ \pm z σ \\ = & 19 \pm 1.96 \times 2.5 \\ = & (14.1, 23.9) \end{array}$

3. $μ = 4.2, σ = 0.25$

$\begin{array}{rcl} C . I & = & μ \pm z σ \\ = & 4.2 \pm 1.96 \times 0.25 \\ = & (3.71, 4.69) \end{array}$

4. $μ = 15, σ = 0.085$

$\begin{array}{rcl} C . I & = & μ \pm z σ \\ = & 15 \pm 1.96 \times 0.085 \\ = & (14.833, 15.167) \end{array}$

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