Question

1. Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, the probability that X is between 48 and 52 is

a)     Approximately 68%

b)     Approximately 95%

c)     Approximately 99%

d)     Can’t tell without the z-table.

2.     A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week. Interpret this value.

a)        Most of the weeks resulted in rat costs of $13.00.

b)        The median cost for the distribution of rat costs is $13.00.

c)        The expected or average cost for all weekly rat purchases is $13.00.

d)        The rat cost that occurs more often than any other is $13.00.

3.     Which of the following about the normal distribution is NOT true?

a)        Theoretically, the mean, median, and mode are the same.

b)        About 2/3 of the observations fall within 1 standard deviation from the mean.

c)        It is a discrete probability distribution.

d)        Its parameters are the mean, μ, and standard deviation, σ.

4.     Suppose a standard normal distributed random variable z turned out to has a value of .75, this means that

a)        25% of the time this random variable is greater than .75.

b)        22.7% of the time this random variable is greater than .75.

c)        75% of the time this random variable is greater than .75.

d)        None of the above is true.

Answer 1

Solution)

1)

a. Approximately 68%

Given , µ = 50 and σ = 2

We know z = (x - µ) /σ

Therefore, P(48 ≤ X ≤ 52) =

48 – 50 52 – 50 PG << 2 2

= P(-1 ≤ Z ≤ 1)

= P(Z ≤ 1)−P(Z ≤ −1)

= 0.8413 - 0.1587 (Using the areas under the standard normal curve table)

=0.6827 ~ 68%

This approximately 68% is the answer.

2)

c. The expected or average costs for all weekly rat purchases is $13.00

Step-by-step explanation:

a. This is incorrect since individual values don't interfere in the mean. For instance, if half the weeks had a cost of $12.00 and the other half had a cost of $14.00, the mean cost would still be $13.00 even though no particular week had a $13.00 cost

b. Incorrect. The median is the central value in a distribution; the median and the mean are not necessarily the same.

c. A mean value of $13.00 means that over a period of 52 weeks, the company can expect to spend $13.00 per week on rat purchases.

d. Incorrect, same as item a.

3)

c. It is a discrete probability distribution.

Step-by-step explanation:

a. Since the distribution is symmetrical, mean, median and mode coincide. Thus, Mean = Median = Mode = μ

b. By the empirical rule , we know that 2/3rd falls within 1 standard deviation from the mean.

  

X 68% of the data values + !

c. This is not true because a normal distribution is a continuous probabilty distribution.

d. Every unique pair of mean,μ , and standard deviation, σ describes a different normal distribution. Hence true.

4)

b.0.226622.7% of the time this random variable is greater than .75.

Using the z- table -

If z-score is 0.75, then 0.77337 of the time the random variable lies less than 0.75. Therefore,

(1-0.77337) = 0.22663 of the time the random variable lies greater than 0.75.