Question

Researchers developed a "dog IQ test" and find that the population of dogs in the United States has a mean score of µ = 20 on this test with a standard deviation of σ = 2. You decide to test a sample of n = 40 dogs to do some research about whether some breeds are smarter.

Use that information to answer all of the following questions:

A. What is the expected value of the sampling distribution of the mean?

B.  What is the doggie IQ range for the middle 50% of all dogs? (Hint: find the Z-score with a probability closest to 0.25, because you want 25% above the mean and 25% below the mean to get the 'middle 50%', and then use that to find the IQ that matches.)

C. : What is the standard error of the mean going to be when you collect your sample of n = 40?

D.

You decide to first test your own dog (or your closest friend's or family member's, if you don't have one).

What is the probability that your dog will score an "dog IQ" great than 25 (which is the cut-off for "Doggie Genius")?

You collect your sample of n = 40 dogs and find their mean dog IQ is M = 20.5.

E: What is the probability of this sample? That is, what is p(M>20.5)?

You decide to collect another sample of n = 20, where all of the dogs are poodles, because you suspect maybe poodles aren't so smart.

The mean IQ of your sample of poodles is M = 19.

F: What is the probability of this sample? That is, what is p(M<19)? (Hint: remember the sample size is now n = 20.)

G; Recall that Z-scores beyond ±2 are considered "extreme". So, if a family has n = 4 dogs, what would their mean IQ have to be, in order for them to be considered extremely smart?

Answer 1

A.

Expected value of the sampling distribution of the mean = µ = 20

B.

For the middle 50% of all dogs, the percentiles are,

(1 - 0.5)/2 and 1 - (1- 0.5)/2

0.25 and 0.75

Z score for p = 0.25 and 0.75 is  0.6745

Lower IQ limit = 20 - 0.6745 * 2 = 18.651

Upper IQ limit = 20 + 0.6745 * 2 = 21.349

The IQ range for the middle 50% of all dogs is (18.651 ,  21.349)

C.

standard error of the mean = σ  / = 2 / = 0.3162278

D.

X ~ Normal(20, 2)

P(X > 25) = P[Z > (25 - 20)/2] = P[Z > 2.5] =  0.0062

E.

M ~ Normal(20, 0.3162278)

p(M>20.5) = P[Z > (20.5 - 20)/0.3162278] = P[Z > 1.58] =  0.0571

F.

standard error of the mean = σ  / = 2 / = 0.4472136

M ~ Normal(20, 0.4472136)

p(M<19) = P[Z < (19 - 20)/0.4472136] = P[Z < -2.24] =  0.0125

G.

standard error of the mean = σ  / = 2 / = 1

Mean IQ to be considered extremely smart = 20 +  2 * 1 = 22