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gizmo_the_mogwai
3 months ago
12

The average sale price (online) for four year old Harley Davidson touring motorcycles is approximately normally distributed with

a mean of $14,000 and a standard deviation of $4,000.
A) Molly has saved up $15,000 to spend on a motorcycle. What proportion of the available motorcycles of this type can she afford?




B) what is the 30th percentile for the prices of motorcycles of this type?




C) show that a motorcycle priced at $25,000 would be considered an outlier by the 1.5 x IQR rule.

Advanced Placement (AP)
1 answer:
vovikov84 [265]3 months ago
6 0

Answer:

A) Molly is able to afford 60% of the motorcycles available.

B) The price at the 30th percentile is $11,920.

C) A motorcycle priced at $25,000 is an outlier because it exceeds $24,792.

Explanation:

Given data:

Mean (μ) = $14,000

Standard deviation (σ) = $4,000.

A) We calculate the standard score Z for x = $15,000 using the standard normal table (see attached figure):

Z = (x - μ) / σ

Z = (15000 - 14000) / 4000

Z = 0.25

Referring to the table, this corresponds to 60%, meaning Molly can buy 60% of the motorcycles.

B) According to the second figure, the 30th percentile Z-value is -0.52, which translates to a price of:

x = μ + Z × σ

x = 14000 + (-0.52 × 4000)

x = $11,920

C) Using the 1.5 × IQR rule, an outlier is any value larger than 1.5 times the interquartile range added to the third quartile.

The third quartile corresponds to Z = 0.6745:

x = 14000 + 0.6745 × 4000 = $16,698

The first quartile corresponds to Z = -0.6745:

x = 14000 - 0.6745 × 4000 = $11,302

Therefore,

IQR = 16698 - 11302 = $5,396

The cutoff for outliers is:

16698 + 1.5 × 5396 = $24,792

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