Question

Golfer Lexi Thompson is one of the top golfers on the LPGA Tour. The distribution of scores for each of the more than 700 rounds over her LPGA career is approximately normal with a mean of about μ = 70.6 strokes and a standard deviation of about σ = 3.2 strokes. What score is at the 16th percentile of the distribution? Justify your answer.

a. 66.2 strokes

b. 68.8 strokes

c. 71.8 strokes

d. 74.4 strokes

Answer 1

The score at the 16th percentile of the distribution is approximately 68.8 strokes.The correct option is B. 68.8 strokes.

Step-by-step explanation:

To find the score at the 16th percentile, we can use the Z-score formula:

`\[ Z = \frac{{X - \mu}}{{\sigma}} \]`

where are X is the score,
`\( \mu \)`is the mean, and
`\( \sigma \)`is the standard deviation. The Z-score represents the number of standard deviations a data point is from the mean in a normal distribution.

The percentile can be converted to a Z-score using the standard normal distribution table. For the 16th percentile, the Z-score is approximately -0.9945.

Now, rearranging the Z-score formula to solve for X :

`\[ X = Z \cdot \sigma + \mu \]`

Substituting the values, we get:

`\[ X = (-0.9945) \cdot 3.2 + 70.6 \]`

`\[ X \approx 68.8 \]`

Therefore, the score at the 16th percentile is approximately 68.8 strokes.

This means that about 16% of Lexi Thompson's scores fall below 68.8 strokes in her LPGA career. This process involves standardizing the scores to compare them on a common scale, making it easier to interpret their relative positions in the distribution.

The Z-score and percentile calculations are fundamental concepts in statistics, providing a standardized way to assess where a particular data point stands in a distribution. Understanding these concepts is crucial for analyzing and interpreting data in various fields.

The correct option is B. 68.8 strokes.