High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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

Source: www.fueleconomy.gov

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

(b) Determine the quartiles.

(c) Compute and interpret the interquartile range (IQR).

(d) Determine the lower and upper fences. Are there any outliers?

Answer :

Final answer:

The z-score for the individual who obtained 36.3 miles per gallon is approximately -0.3509, indicating it is slightly below the mean. The quartiles are Q1 = 37.4 and Q3 = 40.5, and the interquartile range (IQR) is 3.1 miles per gallon. The lower fence is 32.9 and the upper fence is 45.4, but there are no outliers in the given data set.

Explanation:

(a) To compute the z-score for an individual who obtained 36.3 miles per gallon, we need to calculate the mean and standard deviation for the given data set. The mean of the data set is 37.6667 (rounded to 4 decimal places) and the standard deviation is 3.8835 (rounded to 4 decimal places). The z-score can be calculated using the formula: z = (x - mean) / standard deviation. Plugging in the values, we get: z = (36.3 - 37.6667) / 3.8835 = -0.3509 (rounded to 4 decimal places).

The interpretation of the z-score is that the individual who obtained 36.3 miles per gallon is approximately 0.3509 standard deviations below the mean. This means that their miles per gallon value is slightly below average when compared to the rest of the data set.

(b) To determine the quartiles, we need to arrange the data set in ascending order. After arranging the data set, the first quartile (Q1) can be found by taking the average of the values at the 25th and 26th position, which gives us Q1 = 37.4 (rounded to 1 decimal place). The third quartile (Q3) can be found by taking the average of the values at the 19th and 20th position, which gives us Q3 = 40.5 (rounded to 1 decimal place).

(c) The interquartile range (IQR) is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). So, IQR = Q3 - Q1 = 40.5 - 37.4 = 3.1 (rounded to 1 decimal place). The interpretation of the IQR is that the middle 50% of the data values (from Q1 to Q3) have a range of 3.1 miles per gallon.

(d) The lower fence and upper fence can be calculated using the following formulas:

Lower fence = Q1 - 1.5 * IQR = 37.4 - 1.5 * 3.1 = 32.9 (rounded to 1 decimal place)

Upper fence = Q3 + 1.5 * IQR = 40.5 + 1.5 * 3.1 = 45.4 (rounded to 1 decimal place)

The lower fence represents the lowest acceptable value for a data point before it is considered an outlier, while the upper fence represents the highest acceptable value. Based on the given data set, there are no outliers as all the individual values lie within the range defined by the lower and upper fences.

Learn more about Z-Score, Quartiles, IQR here:

https://brainly.com/question/31528264

#SPJ11

Other Questions