A research firm conducted a survey to determine the mean amount Americans spend on coffee during a week. They found the distribution of weekly spending followed the normal distribution with a population standard deviation of $5. A sample of 49 Americans revealed that x = $20.

What is the point estimate of the population mean?

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Solution to the problem

Let X the random variable that represent the weekly spending of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu,5)[/tex]

Where [tex]\mu[/tex] and [tex]\sigma=5[/tex]

For this case we select a sample of n =49 observations and we got a sample mean os :

[tex]\bar X= 20[/tex]

From the definition of mean we have that :

[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

And if we find the expected value of this estimator we got this:

[tex] E(\bar X) = E( \frac{\sum_{i=1}^n X_i}{n}) = \frac{1}{n} n \mu = \mu[/tex]

So then the best estimator unbiased for the population mean is the sample mean:

[tex]\hat \mu = \bar X = 20[/tex]

joaobezerra joaobezerra · Answer 2 · 2020-04-16T02:14:46+02:00

Answer:

By the Central Limit Theorem, the point estimate of the population mean is $20.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

The mean of the sample is $20.

So, by the Central Limit Theorem, the point estimate of the population mean is $20.