The city fuel economy of a 2009 Toyota 4 Runner 2WD 6 cylinder 4 L automatic 5 -speed using reg

step 1 Problem 9 deals with fuel efficiency of cars and trucks, and the question gives us a sample size

The city fuel economy of a 2009 Toyota 4 Runner 2WD 6...

The city fuel economy of a 2009 Toyota 4 Runner 2WD 6 cylinder $4 \mathrm{~L}$ automatic 5 -speed using regular gas is a normally distributed random variable with a range 16 MPG to 21 MPG. (a) Estimate the standard deviation using Method 3 (the Empirical Rule for a normal distribution). (b) What sample size is needed to estimate the mean with 90 percent confidence and an error of $\pm 0.25$ MPG?

The city fuel economy of a 2009 Toyota 4 Runner 2WD 6 cylinder $4 \mathrm{~L}$ automatic 5 -speed using regular gas is a normally distributed random variable with a range 16 MPG to 21 MPG. (a) Estimate the standard deviation using Method 3 (the Empirical Rule for a normal distribution).
(b) What sample size is needed to estimate the mean with 90 percent confidence and an error of $\pm 0.25$ MPG?

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve. It is defined by its mean and standard deviation, where the mean determines the center of the distribution and the standard deviation measures the spread of the data. Many natural phenomena, including measurement errors and test scores, are assumed to be normally distributed, making it a fundamental concept in statistics.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in making quick estimates about the spread of the data and is especially useful when the precise parameters of the distribution are unknown.

Standard Deviation Estimation

Standard deviation estimation involves determining the spread or dispersion of data points in a dataset relative to the mean. One common approach when dealing with normally distributed data is to use the range of the data along with the Empirical Rule, particularly when the data range corresponds to a specific multiple of the standard deviation. This method provides a practical way of inferring variability when only range information is available.

Confidence Interval for the Mean

A confidence interval provides a range of values in which the true population mean is likely to lie with a certain level of confidence. It is calculated based on the sample mean, the estimated standard deviation, and a critical value (typically a z-value from the normal distribution) corresponding to the desired confidence level. The confidence interval helps quantify the uncertainty associated with the estimation process.

Sample Size Determination

Sample size determination is the process of calculating the number of observations needed to estimate a population parameter with a desired level of accuracy and confidence. In the context of estimating a mean, the sample size is influenced by the required margin of error, the variability of the data (i.e., the standard deviation), and the critical value corresponding to the chosen confidence level. It is a critical concept in designing studies to ensure that statistical estimates are both precise and reliable.

Transcript

Please give Ace some feedback