Chapter 5 Using the distribution of \(\overline{X}\) to answer questions

Once we know the distribution, or approximate distribution, of the sample mean, we can use that distribution to answer some questions. We can do this by applying what we learnt in the previous topic, when we learnt how to use the Normal distribution to calculate probabilities.

Consider the following. Suppose the cholesterol of a population of people is normally distributed with a mean of \(\mu = 5\) and variance of \(\sigma^2\) = 1. If we let \(\overline{X}\) be the mean of a random sample of \(n = 50\) people, we can write down the distribution of \(\overline{X}\) as follows: \[\overline{X} \sim N(5, \frac{1}{50})\]

Knowing the distribution of \(\overline{X}\), we can visualise its distribution:

With this distribution, we can now ask questions such as:

If the true mean of this population is \(\mu = 5\), what is the probability that when we take a sample of \(n = 50\) people, the sample mean would be 4.8 or less. That is, what is \(P(\overline{X} \leq 4.8)\)?
If the true mean of this population is \(\mu = 5\), what is the probability that when we take a sample of \(n = 50\) people, the sample mean would be between 4.8 and 5.2? That is, what is \(P(4.8 \leq \overline{X} \leq 5.2)\)?
If the true mean of this population is \(\mu = 5\), what is the probability that when we take a sample of \(n = 50\) people, the sample mean would be 5.2 or more? That is, what is \(P(\overline{X} \geq 5.2)\)?
If the true mean of this population is \(\mu = 5\), what is the probability that when we take a sample of \(n = 50\) people, the sample mean would be less than 4.8 OR more than 5.2? That is, what is \(P(\overline{X} \leq 4.8) + P(\overline{X} \geq 5.2)\)?

These probabilities are represented below:

Your turn:

Given \(\overline{X} \sim N(5, \frac{1}{50})\), calculate the following probabilities. Round your answers to four decimal places. (Note: you will need to use R or jamovi to answer these questions. For each question, we would suggest drawing a picture of the Normal curve with the probability represented - see examples above - to help you answer the question)

\(P(\overline{X} \leq 5.3)\)?

Try round(pnorm(5.3, 5, sqrt(1/50)), 4)

\(P(4.7 \leq \overline{X} \leq 5.3)\)?
\(P(\overline{X} \geq 5.3)\)?
\(P(\overline{X} \leq 4.7) + P(\overline{X} \geq 5.3)\)?

Using the below code, we calculate the answer as 0.9831
```
round(pnorm(5.3, 5, sqrt(1/50)), 4)
```

Using the below code, we calculate the answer as 0.9661

round(pnorm(5.3, 5, sqrt(1/50)) - pnorm(4.7, 5, sqrt(1/50)), 4)

Using the below code, and by using the complement rule, we calculate the answer as 0.0169
```
round(1 - pnorm(5.3, 5, sqrt(1/50)), 4)
```

Using the below code, and by symmetry, we calculate the answer as 0.0339

round(2*pnorm(4.7, 5, sqrt(1/50)), 4)

Alternatively, the below code produces the same result:

round(pnorm(4.7, 5, sqrt(1/50)) + (1 - pnorm(5.3, 5, sqrt(1/50))), 4)

[1] 0.0339