A population has a normal distribution with a mean of 32.9 and a standard deviation of 35.2. Draw4.a normal curve with labels, shading, and what you entered into the calculator for full credit. Roundyour answer to 4 decimal places.Find the probability that a randomly selected item is between 27.5 and 40.6a)Find the probability that a sample size of 25 is randomly selected and is less than 28.2.b)

Question
Asked Oct 24, 2019
A population has a normal distribution with a mean of 32.9 and a standard deviation of 35.2. Draw
4.
a normal curve with labels, shading, and what you entered into the calculator for full credit. Round
your answer to 4 decimal places.
Find the probability that a randomly selected item is between 27.5 and 40.6
a)
Find the probability that a sample size of 25 is randomly selected and is less than 28.2.
b)
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A population has a normal distribution with a mean of 32.9 and a standard deviation of 35.2. Draw 4. a normal curve with labels, shading, and what you entered into the calculator for full credit. Round your answer to 4 decimal places. Find the probability that a randomly selected item is between 27.5 and 40.6 a) Find the probability that a sample size of 25 is randomly selected and is less than 28.2. b)

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Expert Answer

Step 1

a)

The probability that a randomly selected item is between 27.5 and 40.6 is obtained below:

The value of mean μ = 32.9 and standard deviation is 35.2

The following normal curve corresponds to P (27.5 ≤ X ≤ 40.6) is obtained below:

The P (27.5 ≤ X ≤ 40.6) is developed using MINITAB:

  • Choose Graph > Probability Distribution Plot choose View Probability > OK.
  • From Distribution, choose ‘Normal’ distribution.
  • In Mean enter the value as 32.9.
  • In Standard deviation enter the value as 35.2.
  • Click the Shaded Area
  • Choose X Value and Middle for the region of the curve to shade.
  • Enter the X value 1 as 27.5.
  • Enter the X value 2 as 40.6.
  • Click OK.

Follow the above instructions to obtain the MINITAB output.

Distribution Plot
Normal, Mean-32.9, StDev-35.2
0.012
0.1475
0.010
0.008
0.006
0.004
0.002
0.000
27.5 32.9 40.6
Density
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Distribution Plot Normal, Mean-32.9, StDev-35.2 0.012 0.1475 0.010 0.008 0.006 0.004 0.002 0.000 27.5 32.9 40.6 Density

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Step 2

From the MINITAB output the value of P (27.5 ≤ X ≤ 40.6) is 0.1475.

(b)

The probability that a sample size 25 is randomly selected and is less than 28.2 is obtained below:

By central limit theorem, the sample size is 25 with mean μ = 32.9 and standard deviation, 35.2 /sqrt (25) = 7.04.

The following normal curve corresponds to P (X < 28.2) is obtained below:

The P (X < 28.2) is developed using MINITAB:

  • Choose Graph > Probability Distribution Plot choose View Probability > OK.
  • From Distribution, choose ‘Normal’ distribution.
  • In Mean enter the va...
Distribution Plot
Normal, Mean 32.9, StDev 7.04
0.06
0.05
0.04
0.03
0.02
0.2522
0.01
0.00
28.2 32.9
Density
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Distribution Plot Normal, Mean 32.9, StDev 7.04 0.06 0.05 0.04 0.03 0.02 0.2522 0.01 0.00 28.2 32.9 Density

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