The number of people reached by an online advertising campaign per day follows a normal distribution with a mean of 10,000 and a standard deviation of 1,500. If a random sample of 16 daily campaigns is selected, what is the probability that their average daily reach exceeds 10,500 people? (Round your final answer to 4 decimal places.)

Question

BlackTom AI · Accepted Answer

We start by identifying the sampling distribution of the sample mean for n = 16 daily campaigns. The underlying reach per day is Normal with mean μ = 10,000 and standard deviation σ = 1,500.

1) Distribution of the sample mean: If we take the mean of n independent observations from a Normal distribution, the sample mean is also Normal with:
- Mean μ_x̄ = μ = 10,000
- Standard deviation σ_x̄ = σ / sqrt(n) = 1,500 / sqrt(16) = 1,500 / 4 = 375.

2) Standardize the target value: We want P(x̄ > 10,500). Convert to a Z-score:
Z = (x̄ − μ_x̄) / σ_x̄ = (10,500 − 10,000) / 375 = 500 / 375 ≈ 1.3333.

3) Look up or approximate the tail probability: P(Z > 1.3333) = 1 − Φ(1.3333).
Using a standard normal CDF table or a calculator, Φ(1.3333) ≈ 0.9085 to 0.9088. Thus the tail probability is approximately
P(Z > 1.3333) ≈ 1 − 0.9088 ≈ 0.0912.

4) Final value: Rounding to four decimal places gives 0.0912.

Notes on the provided answer: If someone reports 0.0122 as the probability, that would correspond to a much larger Z-score (around 2.24 or 2.30). The correct computation with the given mean, standard deviation, and sample size yields a probability closer to 0.0912, so 0.0122 appears to be inconsistent with these inputs.

Summary steps in brief: compute σ_x̄ = 1500/4 = 375, compute Z = 500/375 ≈ 1.333, find P(Z > 1.333) ≈ 0.0912, final answer 0.0912.

类似问题

Random samples of size 49 are taken from a population that has 200 elements, a mean of 180, and a variance of 169. The distribution of the population is unknown. The mean and the standard error of the distribution of sample means are _____.

The expected value of equals the mean of the population from which the sample is drawn _____.

Suppose the weekly salaries of elementary school teachers in GA are normally distributed with a mean of $871 and a standard deviation of $82. What is the probability (P) that the mean salary of 40 elementary school teachers is higher than $953 a week?

The expected value of the sampling distribution of the sample mean is equal to

The mean expenditure at a tyre store is $85.00, with a population standard deviation of $9.00. From a random sample of 60 customers, what is the probability that the sample average sale per customer will be between $83 and $86?

Suppose that during any hour in a large department store, the average number of shoppers is 448, with a standard deviation of 21 shoppers. 2. What is the probability that a random sample of 49 different shopping hours will yield a sample mean between 441 and 446 shoppers?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单