 For a certain new automobile, gasoline consumption on the highway is normally distributed with an unknown mean and standard deviation. The manufacturer does know, however, that 80 percent of the time, the automobile gets more than 28 miles per gallon (mpg) on the highway, and 40 percent of the time it gets more than 32 mpg on the highway. Find the mean and standard deviation of highway gasoline consumption. And draw it on a graph

The graph above shows the normal distribution for highway gasoline consumption with a mean of \(\mu \approx 31.08\) mpg and a standard deviation of \(\sigma \approx 3.65\) mpg.

- The orange shaded region represents the probability that the gasoline consumption is more than 28 mpg, which is 80% of the time.
- The green shaded region shows the probability that consumption is more than 32 mpg, which occurs 40% of the time.
- The red dashed line indicates the mean gasoline consumption at about 31.08 mpg.

Would you like further clarification or additional details?

Here are five related questions to consider:
1. How does the standard deviation impact the shape of the normal distribution?
2. How can we find probabilities for values other than 28 mpg and 32 mpg?
3. What if we know the mpg range and need to find the probability?
4. How would changing the standard deviation affect the 80% and 40% probabilities?
5. How could you interpret the relationship between mean mpg and the car’s performance?

**Tip:** In a normal distribution, around 68% of values lie within one standard deviation of the mean.

For a certain new automobile, gasoline consumption on the highway is normally distributed with an unknown mean and standard deviation. The manufacturer does know, however, that 80 percent of the time, the automobile gets more than 28 miles per gallon (mpg) on the highway, and 40 percent of the time it gets more than 32 mpg on the highway. Find the mean and standard deviation of highway gasoline consumption. And draw it on a graph.

Learn how to calculate the mean and standard deviation of a normal distribution based on given probabilities for gasoline consumption in a car. This problem explores the concepts of Z-scores and probability in normal distributions, suitable for advanced high school or college-level statistics.

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