document.write( "Question 1199858: Given a test that is normally distributed with a mean of 64 and a standard deviation of
\n" ); document.write( " 13, find:
\n" ); document.write( "i) the probability that a single score drawn at random will be greater than 70.
\n" ); document.write( "ii) the probability that a sample of 25 scores will have a mean less than 60.
\n" ); document.write( "iii) the probability that the mean of a sample of 16 scores will be more than population
\n" ); document.write( "mean by at least 12. \n" ); document.write( "
Algebra.Com's Answer #848291 by CPhill(1959)\"\" \"About 
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**i) Probability of a single score drawn at random being greater than 70:**\r
\n" ); document.write( "\n" ); document.write( "1. **Standardize the value:**
\n" ); document.write( " * z = (X - μ) / σ
\n" ); document.write( " * z = (70 - 64) / 13
\n" ); document.write( " * z = 0.46\r
\n" ); document.write( "\n" ); document.write( "2. **Find the probability using a standard normal distribution table or calculator:**
\n" ); document.write( " * P(X > 70) = P(Z > 0.46)
\n" ); document.write( " * Using a z-table, look up the area to the left of z = 0.46 and subtract it from 1:
\n" ); document.write( " * P(Z > 0.46) = 1 - P(Z ≤ 0.46) ≈ 1 - 0.6772 = 0.3228\r
\n" ); document.write( "\n" ); document.write( " **Therefore, the probability that a single score drawn at random will be greater than 70 is approximately 0.3228.**\r
\n" ); document.write( "\n" ); document.write( "**ii) Probability that a sample of 25 scores will have a mean less than 60:**\r
\n" ); document.write( "\n" ); document.write( "1. **Calculate the standard error of the mean:**
\n" ); document.write( " * σx̄ = σ / √n
\n" ); document.write( " * σx̄ = 13 / √25
\n" ); document.write( " * σx̄ = 13 / 5
\n" ); document.write( " * σx̄ = 2.6\r
\n" ); document.write( "\n" ); document.write( "2. **Standardize the value:**
\n" ); document.write( " * z = (x̄ - μ) / σx̄
\n" ); document.write( " * z = (60 - 64) / 2.6
\n" ); document.write( " * z = -1.54\r
\n" ); document.write( "\n" ); document.write( "3. **Find the probability using a standard normal distribution table or calculator:**
\n" ); document.write( " * P(x̄ < 60) = P(Z < -1.54)
\n" ); document.write( " * From the z-table, P(Z < -1.54) ≈ 0.0618\r
\n" ); document.write( "\n" ); document.write( " **Therefore, the probability that a sample of 25 scores will have a mean less than 60 is approximately 0.0618.**\r
\n" ); document.write( "\n" ); document.write( "**iii) Probability that the mean of a sample of 16 scores will be more than the population mean by at least 12:**\r
\n" ); document.write( "\n" ); document.write( "* **Calculate the standard error of the mean:**
\n" ); document.write( " * σx̄ = σ / √n
\n" ); document.write( " * σx̄ = 13 / √16
\n" ); document.write( " * σx̄ = 13 / 4
\n" ); document.write( " * σx̄ = 3.25\r
\n" ); document.write( "\n" ); document.write( "* **Determine the desired sample mean:**
\n" ); document.write( " * Sample Mean (x̄) = Population Mean (μ) + Difference
\n" ); document.write( " * x̄ = 64 + 12 = 76\r
\n" ); document.write( "\n" ); document.write( "* **Standardize the value:**
\n" ); document.write( " * z = (x̄ - μ) / σx̄
\n" ); document.write( " * z = (76 - 64) / 3.25
\n" ); document.write( " * z = 3.69\r
\n" ); document.write( "\n" ); document.write( "* **Find the probability using a standard normal distribution table or calculator:**
\n" ); document.write( " * P(x̄ > 76) = P(Z > 3.69)
\n" ); document.write( " * From the z-table, P(Z > 3.69) is very close to 0.\r
\n" ); document.write( "\n" ); document.write( " **Therefore, the probability that the mean of a sample of 16 scores will be more than the population mean by at least 12 is extremely small (approximately 0).**
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