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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.

Answer :

Answer:

A) Molly can buy 60% of the available motorcycles.

B) The 30th percentile is $11920

C) $25,000 is considered an outlier because is greater than: $24792

Explanation:

Data:

mean, μ =  $14,000

standard deviation, σ = $4,000.

A) Using standard normal distribution table (see first figure attached) we have to compute Z (the standard variable) for an x = $15,000 (x is the actual variable) as follows:

Z = (x - μ)/σ

Z = (15000 - 14000)/4000

Z = 0.25

From the table, that represent 60%, that is, Molly can buy 60% of the available motorcycles.

B) From the second figure attached, we can see that 30% of the results (here, the motorcycles prices) correspond to a value of Z = -0.52. That correspond to a price of:

x = μ + Z*σ

x = 14000 + (-0.52)*4000

x = $11920

C) The 1.5 x IQR rule states that any value bigger than 1.5 times the  interquartile range (IQR) added to the third quartile is an outlier.

The third quartile is located at Z = 0.6745, in terms of the actual variable:

x = μ + Z*σ

x = 14000 + 0.6745*4000

x = $16698

The first quartile is located at Z = -0.6745, in terms of the actual variable:

x = μ + Z*σ

x = 14000 - 0.6745*4000

x = $11302

Then:

IQR = 16698 - 11302 = $5396

The third quartile value added to 1.5 x IQR is equal to 16698 + 1.5*5396 = $24792

${teks-lihat-gambar} jbiain
${teks-lihat-gambar} jbiain

The proportion of the available motorcycles that Molly can afford is 60%.

The 30th percentile for the motorcycle is $11920

It should be noted that $25,000 is considered an outlier because it is greater than $24792

Since, mean, μ =  $14,000 and standard deviation, σ = $4,000, the standard variable Z or x = $15,000 will be:

Z = (x - μ)/σ

Z = (15000 - 14000)/4000

Z = 0.25

We then check the table and see that it implies that Molly can buy 60% of the available motorcycles.

The 30th percentile for the prices of motorcycles of this type corresponds to a value of Z = -0.52. This will then be:

x = μ + Z*σ

x = 14000 + (-0.52) × 4000

x = $11920

Lastly, the third quartile value when added to 1.5 x IQR will be equal to:

= 16698 + (1.5 × 5396) = $24792

Therefore, $25,000 is considered an outlier because it is greater than $24792

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