Question

A basketball coach is looking over the possessions per game during last season. Assume that the possessions per game follows an unknown distribution with a mean of 56 points and a standard deviation of 12 points. The basketball coach believes it is unusual to score less than 50 points per game. To test this, she randomly selects 36 games. Use a calculator to find the probability that the sample mean is less than 50 points. Round your answer to three decimal places if necessary.

Answer 1

56

2

.001

Explanation:

The Central Limit Theorem for Means states that the mean of any sampling distribution of the means is equal to the mean of the population distribution. The standard deviation is equal to the standard deviation of the population divided by the square root of the sample size. So, the mean of this sampling distribution of the means with sample size 36 is 56 points and the standard deviation is 1236√=2 points. The z-score for 50 using the formula z=x¯¯¯−μσ is −3.

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-3.0 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

-2.9 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001

-2.8 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

-2.7 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

-2.6 0.005 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

-2.5 0.006 0.006 0.006 0.006 0.006 0.005 0.005 0.005 0.005 0.005

Using the Standard Normal Table, the area to the left of −3 is approximately 0.001. Therefore, the probability that the sample mean will be less than 50 points is approximately 0.001.

Answer 2

The probability that the sample mean is less than 50 points = 0.002

Explanation:

Step(i):-

Given mean of the normal distribution = 56 points

Given standard deviation of the normal distribution = 12 points

Random sample size 'n' = 36 games

Step(ii):-

Let x⁻ be the random variable of normal distribution

Let x⁻ = 50

`Z = (x^(-)-mean )/((S.D)/(√(n) ) )`

`Z = (50-56 )/((12)/(√(36) ) )= -3`

The probability that the sample mean is less than 50 points

P( x⁻≤ 50) = P( Z≤-3)

= 0.5 - P(-3 <z<0)

= 0.5 -P(0<z<3)

= 0.5 - 0.498

= 0.002

Final answer:-

The probability that the sample mean is less than 50 points = 0.002