Normally distributed random variable and probability

In summary, the top-selling Red and Voss tire has a rating of 60000 miles, but the actual distance it can run until wear-out is normally distributed with a mean of 70000 miles and a standard deviation of 5000 miles. The probability of the tire wearing out before 60000 miles is 0.0228, which is 2 standard deviations from the mean. The probability of a tire lasting more than 79000 miles is also 0.0228. However, there may have been issues with rounding on the online homework site.
  • #1
major_maths
30
0

Homework Statement


The top-selling Red and Voss tire is rated 60000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 70000 miles and a standard deviation of 5000 miles.

A: What is the probability that the tire wears out before 60000 miles?
B: What is the probability that a tire lasts more than 79000 miles?

Homework Equations


z = (Y-μ)/σ

The Attempt at a Solution


I plugged the values into the above equation and got z=-2. Looking at the chart in my book, Pr(z=2)=0.0228 and I figured this should work since z is 2 standard deviations from the mean no matter whether it's positive or negative. When I submitted 0.0228 though, the answer was incorrect.
 
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  • #2
major_maths said:

Homework Statement


The top-selling Red and Voss tire is rated 60000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 70000 miles and a standard deviation of 5000 miles.

A: What is the probability that the tire wears out before 60000 miles?
B: What is the probability that a tire lasts more than 79000 miles?

Homework Equations


z = (Y-μ)/σ

The Attempt at a Solution


I plugged the values into the above equation and got z=-2. Looking at the chart in my book, Pr(z=2)=0.0228 and I figured this should work since z is 2 standard deviations from the mean no matter whether it's positive or negative. When I submitted 0.0228 though, the answer was incorrect.

Your answer is correct. Who told you it is incorrect?

RGV
 
  • #3
It's an online homework assignment. I talked to my teacher and he said the site was having issues with the rounding. Thanks for the help!
 

1. What is a normally distributed random variable?

A normally distributed random variable is a type of continuous probability distribution that follows a symmetric bell-shaped curve. It is characterized by its mean, which represents the center of the distribution, and its standard deviation, which determines the spread of the data around the mean. Many natural phenomena, such as human height and test scores, can be described by a normal distribution.

2. What does it mean for a random variable to be normally distributed?

When a random variable is normally distributed, it means that the values it can take on are more likely to be closer to the mean, rather than being extreme values. This is because the majority of the data falls within one standard deviation of the mean, according to the empirical rule. Additionally, the probability of getting a specific value from a normal distribution is represented by the area under the curve, making it easier to interpret compared to other types of distributions.

3. How is the probability of a normally distributed random variable calculated?

The probability of a normally distributed random variable can be calculated using the cumulative distribution function (CDF). This function takes in the value of the variable, as well as the mean and standard deviation of the distribution, and returns the probability of getting a value equal to or less than the given value. Alternatively, the probability can also be calculated by finding the area under the curve using integration techniques.

4. What is the central limit theorem and its relationship to normally distributed random variables?

The central limit theorem states that as the sample size of a population increases, the distribution of sample means will approach a normal distribution, regardless of the underlying distribution of the population. This means that many real-world phenomena can be approximated by a normal distribution, making it a useful tool for statistical analysis and inference.

5. How can normally distributed random variables be used in hypothesis testing?

In hypothesis testing, normally distributed random variables are used to determine the probability of obtaining a certain sample mean or difference in means from a given population. This is done by calculating the z-score, which represents the number of standard deviations a sample mean is from the population mean. By comparing this z-score to a critical value, we can determine if the results are statistically significant and reject or fail to reject the null hypothesis.

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