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A simple random sample of size n=36 is obtained from a population that is skewed right with mu=80 and sigma=24. (a) Describe the sampling distribution of bar(x). (b) What is P(x > 86.8) ? (c) What is P(x <= 70.2) ? (d) What is P(74 < bar(x) < 89.6) ?

A simple random sample of size n=36 n=36 is obtained from a population that is skewed right with μ=80 \mu=80 and σ=24 \sigma=24 .\newline(a) Describe the sampling distribution of xˉ \bar{x} .\newline(b) What is P(x>86.8) P(x>86.8) ?\newline(c) What is P(x≤70.2) P(x \leq 70.2) ?\newline(d) What is P(74<xˉ<89.6) P(74<\bar{x}<89.6) ?

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Q. A simple random sample of size n=36 n=36 is obtained from a population that is skewed right with μ=80 \mu=80 and σ=24 \sigma=24 .\newline(a) Describe the sampling distribution of xˉ \bar{x} .\newline(b) What is P(x>86.8) P(x>86.8) ?\newline(c) What is P(x≤70.2) P(x \leq 70.2) ?\newline(d) What is P(74<xˉ<89.6) P(74<\bar{x}<89.6) ?
  1. Identify Distribution: Identify the distribution of the sample mean xˉ\bar{x} for a large sample size from a skewed population.\newlineSince n=36n=36 is sufficiently large, by the Central Limit Theorem, xˉ\bar{x} is approximately normally distributed with mean μ=80\mu=80 and standard deviation σ/n=24/36=4\sigma/\sqrt{n}=24/\sqrt{36}=4.
  2. Calculate P(xˉ>86.8)P(\bar{x} > 86.8): Calculate P(xˉ>86.8)P(\bar{x} > 86.8) using the standard normal distribution.\newlineConvert xˉ>86.8\bar{x} > 86.8 to a Z-score: Z=86.8−804=1.7Z = \frac{86.8 - 80}{4} = 1.7.\newlineUsing standard normal tables or calculator, P(Z>1.7)≈0.0446P(Z > 1.7) \approx 0.0446.
  3. Calculate P(xˉ≤70.2)P(\bar{x} \leq 70.2): Calculate P(xˉ≤70.2)P(\bar{x} \leq 70.2) using the standard normal distribution.\newlineConvert xˉ≤70.2\bar{x} \leq 70.2 to a Z-score: Z=70.2−804=−2.45Z = \frac{70.2 - 80}{4} = -2.45.\newlineUsing standard normal tables or calculator, P(Z≤−2.45)≈0.0071P(Z \leq -2.45) \approx 0.0071.
  4. Calculate P(74<xˉ<89.6)P(74 < \bar{x} < 89.6): Calculate P(74<xˉ<89.6)P(74 < \bar{x} < 89.6) using the standard normal distribution.\newlineConvert to Z-scores: Z1=74−804=−1.5Z_1 = \frac{74 - 80}{4} = -1.5 and Z2=89.6−804=2.4Z_2 = \frac{89.6 - 80}{4} = 2.4.\newlineUsing standard normal tables or calculator, P(−1.5<Z<2.4)≈P(Z<2.4)−P(Z<−1.5)≈0.9918−0.0668=0.925P(-1.5 < Z < 2.4) \approx P(Z < 2.4) - P(Z < -1.5) \approx 0.9918 - 0.0668 = 0.925.

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