Suppose that Dunlop Tire manufactures a brand of tires with lifetime miles that follows a normal distribution with mean of 70,000 miles and a standard deviation of 4400 miles.State the distribution of X using the notation from the formula sheet. 1. 2. 3.Determine the percentage of tires that will last at least 75,000 miles?Hint: find P(X ≥ 75000)What percentage of the tires will last 60,000 miles or less? Suppose that Dunlop wants to warrant no more than 2% of its tires. What mileage should the company advertise as its warranty mileage? Hints: Find X using the backwards method…is the X we are looking for less than or more than the mean? Think about the mean tire mileage and compare that to completely worn out tires…then decide where on the curve to plot X. # Probability Distribution Formulas## 1. The Binomial Distribution**Formula and Parameters:**- $ X \sim \text{bin}(n, p) $**TI-84 Distribution Functions:**- $ P(X = x) \iff \text{binpdf}(n, p, x) $- $ P(X \leq x) \iff \text{bincdf}(n, p, x) $**Calculations:**- Expected Value (Mean), $ E(x) = n \cdot p $- Standard Deviation, $ S.D.(x) = \sqrt{n \cdot p \cdot (1-p)} $## 2. The Normal Distribution**Formula and Parameters:**- $ X \sim N(\mu, \sigma) $**Standard Normal Transformation:**- $ Z = \frac{X - \mu}{\sigma} $## 3. The Sample Mean**Formula and Parameters:**- $ \bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}}) $**Standard Normal Transformation:**- $ Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} $## 4. The Sample Proportion**Formula and Parameters:**- $ \hat{P} \sim N\left(P, \sqrt{\frac{P(1-P)}{n}}\right) $**Standard Normal Transformation:**- $ Z = \frac{\hat{P} - P}{\sqrt{\frac{P(1-P)}{n}}} $This information provides an overview of key probability distribution formulas and transformations used in statistical analysis.

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Answered: Suppose that Dunlop Tire manufactures a…

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Suppose that Dunlop Tire manufactures a brand of tires with lifetime miles that follows a normal distribution with mean of 70,000 miles and a standard deviation of 4400 miles.

State the distribution of X using the notation from the formula sheet.

Determine the percentage of tires that will last at least 75,000 miles?

Hint: find P(X ≥ 75000)

What percentage of the tires will last 60,000 miles or less?

Suppose that Dunlop wants to warrant no more than 2% of its tires. What mileage should the company advertise as its warranty mileage? Hints: Find X using the backwards method…is the X we are looking for less than or more than the mean? Think about the mean tire mileage and compare that to completely worn out tires…then decide where on the curve to plot X.

# Probability Distribution Formulas

## 1. The Binomial Distribution

**Formula and Parameters:**
- $ X \sim \text{bin}(n, p) $

**TI-84 Distribution Functions:**
- $ P(X = x) \iff \text{binpdf}(n, p, x) $
- $ P(X \leq x) \iff \text{bincdf}(n, p, x) $

**Calculations:**
- Expected Value (Mean), $ E(x) = n \cdot p $
- Standard Deviation, $ S.D.(x) = \sqrt{n \cdot p \cdot (1-p)} $

## 2. The Normal Distribution

**Formula and Parameters:**
- $ X \sim N(\mu, \sigma) $

**Standard Normal Transformation:**
- $ Z = \frac{X - \mu}{\sigma} $

## 3. The Sample Mean

**Formula and Parameters:**
- $ \bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}}) $

**Standard Normal Transformation:**
- $ Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} $

## 4. The Sample Proportion

**Formula and Parameters:**
- $ \hat{P} \sim N\left(P, \sqrt{\frac{P(1-P)}{n}}\right) $

**Standard Normal Transformation:**
- $ Z = \frac{\hat{P} - P}{\sqrt{\frac{P(1-P)}{n}}} $

This information provides an overview of key probability distribution formulas and transformations used in statistical analysis.

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 + ...

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