Suppose commute times in a large city are normally distributed and that 62.5 % of commuters in t

Suppose commute times in a large city are normally...

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is fully described by two parameters: the mean, which determines its center, and the standard deviation, which determines the spread or dispersion of the data.

Standard Deviation

The standard deviation measures the spread or variability of data values around the mean. In a normal distribution, it indicates how much individual values typically deviate from the mean, influencing the width of the bell curve.

Z-Score Transformation

The z-score transformation standardizes a normal random variable by subtracting the mean and dividing by the standard deviation, converting it to a standard normal variable. This process allows for the use of standard normal distribution tables to determine probabilities and percentiles.

Percentile and Inverse Cumulative Distribution Function (CDF)

Percentiles indicate the relative standing of a value within the distribution, showing the percentage of observations below that value. The inverse CDF, or quantile function, is used to find the specific value (or corresponding z-score) that aligns with a given cumulative probability, which is essential for solving problems involving threshold percentages.

Mean

The mean is the average or central value of a dataset. In the context of normally distributed data, the mean is the point around which the data is symmetrically distributed, and it serves as the balance point of the distribution.

Transcript

00:01 Problem 15, the commute times follows a normal distribution with the standard deviation of 6 .2 minutes.

00:10 If we know that 62 .5 % of these times take more than 21 minutes.

00:21 We want to find the mean.

00:26 To get the mean, let's imagine the normal distribution which is bell -shaped centered at the mean.

00:32 We know that 62 .5 % is greater than 21 minutes which means 21 minutes is to the left of me we have here 21 minutes to make the area under the curve to the right of 21 to be equals 0 .625 because 20 because 62 .5 % is greater than 21 minutes is greater than 21 minutes now we won't find the mean to find the mean we should transform this graph into a z score or a z graph or a standard normal distribution which is centered at 0 and we have a corresponding value to 21 which is z and the area to the right of z equals the same area 0 .625 and to transform from 21 to z we use this formula z equals x minus mu divided by sigma we don't know z but we know x x is 21 minus mu divided by sigma divided by sigma we know sigma is given by 6 .2 this means if we know z we can get mu we can get using the standard normal distribution tables we enter the tables by the area to the left of the z score we can get the area to the left of the z score by subtracting this area from 1 minus 0 .625 because the total area under the normal distribution equals 1 then 1 minus 0 .625 gives 0 .375 we enter the tables to search for the this area and get the corresponding z score we search the area is 0 .375 for this table it gives the area to the left of the z score let's search for 0 .375 we can get there is 0 .34 which means 0 .3 7 here 7 .5 is almost this value this is the closest one which means it corresponds to a value a z value of minus 0 .3 2 because this column represents 0 .01 0 .02 0 .0 .0 .0.

03:42 0 .0...

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