The highway fuel economy of a 2016 Lexus RX 350 FWD 6-cylinder 3.5-L automatic 5-speed using premium fuel is a normally distributed random variable with a mean of u = 26.00 mpg and a standard deviation of o = 2.50 mpg.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Educational Content: Understanding Fuel Economy Through Statistical Analysis**
**Homework: Chapter 8 (Sections 8.4 through 8.6)**
In this exercise, we will explore the highway fuel economy of a 2016 Lexus RX 350 FWD equipped with a 6-cylinder 3.5-L automatic 5-speed engine using premium fuel. The fuel economy is assumed to be a normally distributed random variable characterized by the following parameters: a mean (\(\mu\)) of 26.00 mpg (miles per gallon) and a standard deviation (\(\sigma\)) of 2.50 mpg.
### Part (a): Calculating the Standard Error
The question requires us to determine the standard error of \(\bar{X}\), which is the sample mean from a random sample of 16 fill-ups by a driver. The standard error is calculated using the formula:
\[ \text{Standard Error of } \bar{X} = \frac{\sigma}{\sqrt{n}} \]
- \(\sigma\) (standard deviation of the population) = 2.50 mpg
- \(n\) (sample size) = 16
The standard error can be computed as follows:
\[ \text{Standard Error of } \bar{X} = \frac{2.50}{\sqrt{16}} = \frac{2.50}{4} = 0.6250 \]
### Part (b): Calculating the Confidence Interval
Next, we need to determine the interval within which we expect the sample mean to fall with 90 percent probability. This interval is known as the confidence interval. For a normal distribution, the 90% confidence interval can be found using the z-score corresponding to 0.05 in each tail (since 1 - 0.90 = 0.10, with two tails of 0.05 each).
- The z-value for a 90% confidence interval is approximately ±1.645.
The confidence interval is calculated as follows:
\[ \bar{X} \pm z \cdot \text{Standard Error of } \bar{X} \]
Plugging in the values:
\[ 26.00 \pm 1.645 \times 0.6250 \]
\[ 26.00 \pm 1.0281 \]
Therefore, the 90% confidence interval for the sample mean is:
\[ [24.9719,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9cc7e3d1-0afb-4f50-97d1-c905af49f09d%2F0383d639-c818-4716-97ee-5b5db2ff485d%2Fktx9tss.jpeg&w=3840&q=75)
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