Question

Given a normal distribution with

muμequals=105

and

sigmaσequals=20,

and given you select a sample of

n equals 16n=16​,

complete parts​ (a) through​ (d).Click here to view page 1 of the cumulative standardized normal distribution table.

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Click here to view page 2 of the cumulative standardized normal distribution table.

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a. What is the probability that

Upper X overbarX

is less than

9292​?

​P(Upper X overbarXless than<9292​)equals=. 0047.0047

​(Type an integer or decimal rounded to four decimal places as​ needed.)

b. What is the probability that

Upper X overbarX

is between

9292

and

93.593.5​?

​P(9292less than

​(Type an integer or decimal rounded to four decimal places as​ needed.)

c. What is the probability that

Upper X overbarX

is above

106.6106.6​?

​P(Upper X overbarXgreater than>106.6106.6​)equals=. 3745.3745

​(Type an integer or decimal rounded to four decimal places as​ needed.)

d. There is a

6262​%

chance that

Upper X overbarX

is above what​ value?

Upper X overbarXequals=103.47103.47

​(Type an integer or decimal rounded to two decimal places as​ needed.)

Question is complete. Tap on the red indicators to see incorrect answers.

Answer 1

Using central limit theorem,

P( < x) = P( Z < x - / / sqrt(n) )

a)

P( < 92) = P( Z < 92 - 105 / 20 / sqrt(16) )

= P (Z < -2.6)

= 0.0047

b)

P(92 < < 93.5) = P( < 93.5) - P( < 92)

= P( Z < 93.5 - 105 / 20 / sqrt(16) ) - P( Z < 92 - 105 / 20 / sqrt(16) )

= P( Z < -2.3) - P ( Z < -2.6)

= 0.0107 - 0.0047

= 0.0060

c)

P( > 106.6) = P( Z >106.6 - 105 / 20 / sqrt(16) )

= P(Z > 0.32)

= 0.3745

d)

We have to calculate x such that

P( > x) = 0.62

P( Z > x - / / sqrt(n) ) = 0.62

P( Z < x - / / sqrt(n) ) = 0.38

From the Z table, z-score for the probability of 0.38 is -0.3055

x - / / sqrt(n) = -0.3055

x - 105 / 20 / sqrt(16) = -0.3055

Solve for x

x = 103.47