Question

Use the Empirioal Rule. The mean speed of a sample of vehicles along a stretch of highway is 69 miles per hour, with a standard deviasion of 3 miles per hour, Estimate the percant of vahicles whose speeds are between 60 miles per hour and 78 miles per hour. (Assume the data set has a belishaped distribution.) Approximately K of vehicies tavel between 60 miles por hour and 76 miles per hour.

Answer 1

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to data that follows a normal distribution. It states that:

- Approximately 68% of the data falls within one standard deviation of the mean.

- Approximately 95% of the data falls within two standard deviations of the mean.

- Approximately 99.7% of the data falls within three standard deviations of the mean.

In your case, the mean speed of the vehicles is 69 miles per hour, and the standard deviation is 3 miles per hour. Let's use the Empirical Rule to estimate the percentage of vehicles whose speeds are between 60 miles per hour and 78 miles per hour.

1. Calculate one standard deviation below the mean:

Mean - 1 * Standard Deviation = 69 - 1 * 3 = 66 miles per hour

2. Calculate one standard deviation above the mean:

Mean + 1 * Standard Deviation = 69 + 1 * 3 = 72 miles per hour

So, according to the Empirical Rule, approximately 68% of the vehicles will have speeds between 66 miles per hour and 72 miles per hour.

Now, to estimate the percentage of vehicles whose speeds are between 60 miles per hour and 78 miles per hour, we can consider that this range is wider than two standard deviations below and above the mean (which covers approximately 95% of the data).

So, we can estimate that approximately 95% of the vehicles will have speeds between 66 miles per hour and 72 miles per hour. However, since 78 miles per hour is slightly beyond two standard deviations above the mean, we might expect a bit less than 95% but still close. Let's say around 94% of the vehicles.

Therefore, approximately 94% of the vehicles are estimated to have speeds between 60 miles per hour and 78 miles per hour.

For your second question about the number of vehicles traveling between 60 miles per hour and 76 miles per hour, this range is within two standard deviations below and above the mean. So, you can use the Empirical Rule to estimate that approximately 95% of the vehicles will have speeds between 66 miles per hour and 72 miles per hour (as calculated earlier).

So, approximately 95% of the vehicles are estimated to have speeds between 60 miles per hour and 76 miles per hour.

Explanation: