Question

1. golf drives 2015 ii exercise 28 looked at distances pga golfers can hit the ball. the standard deviation of these average drive distances is 11.2 yards, and the quartiles are q1 = 282.5 yards and q3 = 295.6 yards.

a) write a sentence or two describing the spread in distances based on
i) the quartiles.
ii) the standard deviation.
b) do you have any concerns about using either of these descriptions of spread? explain.
fuel economy the boxplot shows the fuel economy ratings for 67 subcompact cars. some summary statistics are also provided. the extreme outlier is the mitsubishi i - miev, an electric car whose electricity usage is equivalent to 112 miles per gallon.
combined fuel economy
mean stdev min q1 med q3 max n
23.76 11.87 14 20 22 25 112 67
if that electric car is removed from the data set, how will the standard deviation be affected? the iqr?

1. test scores correction after entering the test scores from her statistics class of 25 students, the instructor calculated some statistics of the scores. upon checking, she discovered that she had entered the top score as 46, but it should have been 56.

Explanation:

Step1: Analyze spread using quartiles

The inter - quartile range (IQR) is $IQR = Q_3 - Q_1$. Given $Q_1=282.5$ yards and $Q_3 = 295.6$ yards, so $IQR=295.6 - 282.5=13.1$ yards. This means that the middle 50% of the PGA golfers' average drive distances vary by 13.1 yards.

Step2: Analyze spread using standard deviation

The standard deviation of the average drive distances is 11.2 yards. This indicates that, on average, the individual average drive distances of PGA golfers deviate from the mean by about 11.2 yards.

Step3: Discuss concerns

The standard deviation is sensitive to outliers. If the data has extreme values, the standard deviation may give a misleading picture of the spread. The IQR is resistant to outliers, but it only considers the middle 50% of the data and does not give information about the spread of the entire data set or the tails of the distribution.

Step4: Analyze effect on standard deviation and IQR for fuel - economy data

The electric car (Mitsubishi i - MiEV) is an extreme outlier. The standard deviation is sensitive to outliers. Removing the outlier will decrease the standard deviation as the remaining data points are more closely clustered together. The IQR is resistant to outliers. Removing the outlier will likely have little to no effect on the IQR since the outlier is not part of the middle 50% of the data used to calculate the IQR.

Answer:

a)
i) The middle 50% of PGA golfers' average drive distances vary by 13.1 yards (since $IQR = Q_3 - Q_1=295.6 - 282.5 = 13.1$ yards).
ii) On average, the individual average drive distances of PGA golfers deviate from the mean by about 11.2 yards (standard deviation).
b) The standard deviation is sensitive to outliers and may be misleading if there are extreme values. The IQR only considers the middle 50% of data and does not give information about the entire spread or tails. For the fuel - economy data, removing the outlier (the electric car) will decrease the standard deviation and likely have little to no effect on the IQR.