Question

The mean balance that college students owe on their credit card is $1096 with a standard deviation of $350. If all possible random samples of size 144 are taken from this population, determine the following:

a) name of the Sampling Distribution

b) mean and standard error of the sampling distribution of the mean (use the correct name and symbol for each)

c) percent of sample means for a sample of 144 college students that is greater than $1200

d) probability that sample means for samples of size 144 fall between $900 and $1200.

e) Below which sample mean can we expect to find the lowest 25% of all the sample means?

Answer 1

Mean, = $1096

Standard deviation, = $350

Sample size, n = 144

(a) Name of the sampling distribution: Normal distribution

(From Central Limit Theorem since the sample size is large enough)

(b) Mean of the sampling distribution of the mean,

Standard error of the sampling distribution of the mean,

= 350/√144 = $29.167

(c) The required probability = P{Z > (1200 - 1096)/29.167}

= P(Z > 3.566) = 0.0002 = 0.02%

(d) The required probability

= P{(900 - 1096)/29.167 < Z < (1200 - 1096)/29.167}

= P{-6.72 < Z < 3.566)

≈ 0.9998

(e) Corresponding to lowest 25% of all the sample means, the critical z value is -0.6745

Thus, the required sample mean = $1096 - 0.6745*$29.167

= $1076.33