The following data represent the miles per gallon of a random sample of SMART cars with a three-cylinder, 1.0-liter engine:

31.5, 36.0, 37.8, 38.4, 40.1, 42.3, 34.3, 36.3, 37.9, 38.8, 40.6, 42.7, 34.5, 37.4, 38.0, 39.3, 41.4, 43.5, 35.5, 37.5, 38.3, 39.5, 41.5, 47.5

(a) Compute the z-score corresponding to the individual who obtained 36.3 miles per gallon. Interpret this result.

(b) Determine the quartiles.

(a) The z-score corresponding to the individual who obtained 36.3 miles per gallon is approximately -0.33, indicating that this data point is about 0.33 standard deviations below the mean.

(b) The quartiles for the dataset are as follows: Q1 ≈ 36.75 MPG, Q2 (median) ≈ 38.4 MPG, and Q3 ≈ 40.6 MPG.

Explanation:

(a) To compute the z-score for the data point 36.3 MPG, we use the formula:

[tex]\[Z = \frac{X - \mu}{\sigma},\][/tex]

Where:

Z is the z-score,

X is the data point (36.3 MPG),

μ is the mean of the dataset,

σ is the standard deviation of the dataset.

First, we calculate the mean and standard deviation of the given data:

Mean (μ) = (31.5 + 36.0 + ... + 41.5 + 47.5) / 24 ≈ 38.725 MPG

Standard Deviation (σ) ≈ 4.343 MPG

Now, we can plug these values into the z-score formula:

[tex]\[Z = \frac{36.3 - 38.725}{4.343} ≈ -0.33.\][/tex]

The negative sign indicates that the data point is below the mean. The z-score of -0.33 means it is approximately 0.33 standard deviations below the mean.

(b) To determine the quartiles, first, we need to sort the data in ascending order. The quartiles are the values that divide the data into four equal parts. For this dataset, the quartiles are as follows:

Q1 (25th percentile) ≈ 36.75 MPG

Q2 (50th percentile, the median) ≈ 38.4 MPG

Q3 (75th percentile) ≈ 40.6 MPG

These quartiles help us understand the spread and distribution of the data, with Q2 being the median value.

Learn more about z-score

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