Question

According to Wikipedia, the speed of an adult cheetah varies Normally with a mean of 71.5 mph (miles per hour) and a standard deviation of 3.5 mph.

Hint: For each question ask yourself if the question is about the population information or Sampling distribution of sample means.

QUESTIONS BELOW:

1.What is the probability that a randomly chosen cheetah will run faster than 77.77mph? (Draw a picture with relevant values (x,z,p center, spread) and inflection points identified, and an appropriate label to support your full-sentence answer.)

2.What is the probability that a random selection of 16 cheetahs will have an average speed of 77.77 mph or slower? (Draw a picture with relevant values (x,z,p center, spread) and inflection points identified, and an appropriate label to support your full-sentence answer.)

3.How fast does a cheetah have to run to be in the top 10% of fastest cheetahs? (Draw a picture with relevant values (x,z,p center, spread) and inflection points identified, and an appropriate label to support your full-sentence answer.)

4.What is the probability that a random selection of 25 cheetahs will have an average speed between 65 and 80mph? (Draw a picture with relevant values (x,z,p center, spread) and inflection points identified, and an appropriate label to support your full-sentence answer.)

Answer 1

Let the random variable X denote the speed of an adult cheetah.

Given that X ~ N(71.5, 3.5)

By definition of Z score,

1.The probability that a randomly chosen cheetah will run faster than 77.77mph

P(X > 77.77)

From Standard Normal table,

= 1 - 0.96327

= 0.0367

We find that X = 77.7 i.e Z = 1.79 lies above the inflection point. The probability that a randomly chosen cheetah will run faster than 77.77mph is 0.0367.

2.The probability that a random selection of 16 cheetahs will have an average speed of 77.77 mph or slower

=

(From standard normal table)

We may say that this is a certain event, with the area covering the entire normal curve:

The probability that a random selection of 16 cheetahs will have an average speed of 77.77 mph or slower .

3. To find: 90th percentile (P₉₀):

From Standard Normal table, we may obtain the Z score corresponding to the left tail area of 0.90:

The exact value of Z would be Z = 1.282

= 75.99

A cheetah would have to run at the speed of 75.99 mph to be in the top 10% of fastest cheetahs.

4. The probability that a random selection of 25 cheetahs will have an average speed between 65 and 80mph

From Standard Normal table,

= 0.99245 - 0.03144

= 0.961

The probability that a random selection of 25 cheetahs will have an average speed between 65 and 80mph = 0.961