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