Question

the composite scores of individual students on the act college entrance examination in 2009 followed a normal distribution with mean 21.1 and standard deviation 5.1. (a) what is the probability that a single student randomly chosen from all those taking the test scores 23 or higher? show your work. (b) now take an srs of 50 students who took the test. what is the probability that the mean score x of these students is 23 or higher? show your work.

Answer 1

a) The probability that a single student randomly chosen from all those taking the test scores 23 or higher is 0.3557

b) The probability that the mean score x of these students is 23 or higher is 0.0043

What is Probability?

Probability is synonymous with possibility. It is a mathematical branch that deals with the occurrence of a random event.

Given that ,

mean = \mu = 21.1

standard deviation = \sigma = 5.1

a) P(x ≥ 23 ) = 1 - P(x ≤ 23)

= 1 - P[(x - \mu) / \sigma ≤ (23-21.1) /5.1 ]

= 1 - P(z ≤ 0.37)

= 1 - 0.6443 = 0.3557

Probability= 0.3557

b) n = 50

mu\bar x = \mu = 21.1

σbar x = σ / \$\sqrt{n}$ = 5.1/ \$\sqrt{50}$ = 0.7212

P(\bar x ≥ 23) = 1 - P(\bar x ≤ 23 )

= 1 - P[(\bar x - \mu\bar x ) / \sigma\bar x ≤ (23 - 21.1) / 0.7212 ]

= 1 - P(z ≤ 2.63)

= 1 - 0.9957 = 0.0043

Probability = 0.0043

a) The chance of a single student picked at random from all those taking the test scoring 23 or higher is 0.3557.

b) The chance that the mean x of these students is 23 or greater is 0.0043.

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