The mean selling price of senior condominiums in Green...

The mean selling price of senior condominiums in Green Valley over a year was $$\$ 215,000$$. The population standard deviation was $$\$ 25,000$$. A random sample of 100 new unit sales was obtained. a. What is the probability that the sample mean selling price was more than $$\$ 210,000$$ ? b. What is the probability that the sample mean selling price was between $$\$ 213,000$$ and $$\$ 217,000$$ ? c. What is the probability that the sample mean selling price was between $$\$ 214,000$$ and $$\$ 216,000$$ ? d. Without doing the calculations, state in which of the following ranges the sample mean selling price is most likely to lie: $$ \begin{aligned} & \$ 213,000 \text { to } \$ 215,000 ; \$ 214,000 \text { to } \$ 216,000 \text {; } \\ & \$ 215,000 \text { to } \$ 217,000 ; \$ 216,000 \text { to } \$ 218,000 \end{aligned} $$ e. Suppose that, after you had done these calculations, a friend asserted that the population distribution of selling prices of senior condominiums in Green Valley was almost certainly not normal. How would you respond?

The mean selling price of senior condominiums in Green Valley over a year was $$\$ 215,000$$. The population standard deviation was $$\$ 25,000$$. A random sample of 100 new unit sales was obtained.
a. What is the probability that the sample mean selling price was more than $$\$ 210,000$$ ?
b. What is the probability that the sample mean selling price was between $$\$ 213,000$$ and $$\$ 217,000$$ ?
c. What is the probability that the sample mean selling price was between $$\$ 214,000$$ and $$\$ 216,000$$ ?
d. Without doing the calculations, state in which of the following ranges the sample mean selling price is most likely to lie:
$$
\begin{aligned}
& \$ 213,000 \text { to } \$ 215,000 ; \$ 214,000 \text { to } \$ 216,000 \text {; } \\
& \$ 215,000 \text { to } \$ 217,000 ; \$ 216,000 \text { to } \$ 218,000
\end{aligned}
$$
e. Suppose that, after you had done these calculations, a friend asserted that the population distribution of selling prices of senior condominiums in Green Valley was almost certainly not normal. How would you respond?

Key Concepts

Central Limit Theorem

This theorem states that when independent random samples of a sufficient size are drawn from any population with a finite level of variance, the distribution of the sample mean will tend to be normal regardless of the shape of the original population distribution. It underpins why probabilities related to sample means can often be calculated using the standard normal distribution.

Sampling Distribution of the Mean

This concept refers to the probability distribution of all possible sample means, calculated from all possible samples of a fixed size drawn from a population. It has a mean equal to the population mean and a standard deviation (the standard error) that is smaller than the population standard deviation, making it a crucial concept for estimating probabilities and confidence intervals.

Standard Error

Standard error is the standard deviation of the sampling distribution of the mean. It is computed as the population standard deviation divided by the square root of the sample size, quantifying the degree of variability in the sample mean relative to the true population mean.

Z-Scores and the Standard Normal Distribution

Z-scores are standardized scores that describe how many standard deviations an element is from the mean. In the context of the sampling distribution of the mean, z-scores allow for the conversion of sample means into a standard normal variable so that probabilities can be easily looked up or computed using standard normal tables or software.

Normality Assumption and Large Sample Sizes

Even if the underlying population distribution is not normal, the central limit theorem ensures that the sampling distribution of the mean will be approximately normal for large sample sizes. This justifies the use of normal probability calculations in making inferences about the population mean, even when the original population data may not follow a normal distribution.

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