The average miles driven annually per licensed driver in the...

Name: Problem 15, Chapter 3: Elementary Statistics a Step by Step Approach
Uploaded: 2020-07-21T06:24:47-08:00
Duration: 5 min 31 s
Channel: Wendi Obritz
Description: The average miles driven annually per licensed driver in the United States is approximately 14,090 miles. If we assume a fairly mound-shaped distribution with a standard deviation of approximately 3500 miles, find the following: a. z score for 16,000 miles b. z score for 10,000 miles c. Number of miles corresponding to z scores of 1.6, ?0.5, and 0.

The average miles driven annually per licensed driver in the United States is approximately 14,090 miles. If we assume a fairly mound-shaped distribution with a standard deviation of approximately 3500 miles, find the following: a. z score for 16,000 miles b. z score for 10,000 miles c. Number of miles corresponding to z scores of 1.6, ?0.5, and 0.

The average miles driven annually per licensed driver in the United States is approximately 14,090 miles. If we assume a fairly mound-shaped distribution with a standard deviation of approximately 3500 miles, find the following:
a. z score for 16,000 miles
b. z score for 10,000 miles
c. Number of miles corresponding to z scores of 1.6, ?0.5, and 0.

Key Concepts

Raw Score and Z Score Conversion

This concept involves converting a raw score from a data set into a z score (and vice versa) using the formula z = (raw score - mean) / standard deviation. It provides a method for understanding where a particular observation stands relative to the overall distribution.

Z Score

A z score indicates how many standard deviations a given value is from the mean of its distribution. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z scores are useful in standardizing different distributions, allowing for comparison and calculation of probabilities.

Mean

The mean is the average value of a data set and serves as the central point of the normal distribution. It represents the balance point of the distribution, where the values are evenly distributed around it.

Normal Distribution

The normal distribution is a bell-shaped probability distribution that is characterized by its symmetric shape about the mean. It is frequently used in statistics to describe the distribution of many natural phenomena and is defined by its mean and standard deviation.

Standard Deviation

The standard deviation is a measure of the spread or dispersion of a set of data values around the mean. It quantifies how much the data typically deviates from the mean, and in the context of a normal distribution, it is a key parameter in determining the distribution's shape.

Transcript

00:01 So in this question, we're given some information about the average miles people drive with a standard deviation and ask to find some z scores and evaluate some z scores.

00:12 So here's our given information.

00:14 We are given that x bar is 14 ,090.

00:19 The average amount of miles driven is 14 ,090 with a standard deviation of 3 ,500.

00:27 And it would be good for us to review what the formula is for a z score since we have to find the z score.

00:36 So the formula is x minus x bar over s, where x is individual data value, x bar is the sample mean, and s is the sample standard deviation.

00:48 All right.

00:48 So question a says, what is the z score if somebody drives on average 16 ,000 miles? so 16 ,000 is x.

01:02 Here's our mean and our standard deviation.

01:05 We simply have to plug these values into our formula and divide.

01:17 So when we do that calculation, 16 ,000 minus 1490 divided by 3 ,500 is 0 .5, well, round to two decimal places, 0 .55.

01:33 So somebody who drives 16 ,000 is driving over the national average.

01:39 So their z score is positive, but it's not a whole standard deviation over the average.

01:44 It's just a little.

01:45 So just 0 .55 standard deviations above average.

01:50 Question b says, what is the z score for somebody who drives? oops, not x bar, x.

01:59 For somebody who drives 10 ,000 miles.

02:01 So somebody who drives less than average.

02:06 Let me just clean up that notation.

02:09 We're calculating a z score.

02:11 All right.

02:11 So z equals then 10 ,000 minus 14 ,090, divided by 3 ,500...

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