In Topic 4, we learnt about sampling distributions. In this computer lab, we will consider different calculations relating to the Central Limit Theorem and the distribution of the sample mean.
In Lecture 4, we covered material that will be very helpful for today’s computer lab. If you have not already watched Lecture 4, you may wish to do so now.
After working through the questions in this computer lab, you will be ready to complete Quiz 5. If you have time during today’s lab, you may like to work on the quiz.
Important: For all the questions in this lab, where relevant, please round your answers to 4 decimal places of accuracy. When using the Rj Editor, remember to do the following:
options(digits = 6) to control the level of accuracy displayed in the Rj Editor results window.If you need a refresher on the above, please watch the following video before proceeding.
🏡 Imagine that the year is 2040, and you are running an inventory on deep-space mining craft for your company. You know that the number of fuel tanks on all the different crafts are normally distributed, with population mean \(\mu = 12\) fuel tanks per craft, and population variance \(\sigma^2 = 34.42\).
Suppose that you select a random sample of \(n=25\) spacecraft in the shipyard to assess.
🏡 Let \(\overline{X}\) denote the sample mean. Write down the distribution of \(\overline{X}\) in mathematical notation.
💻 Using this distribution, carry out the following calculations in jamovi. Remember, it can be helpful to sketch pictures of the distribution when working out these different probabilities.
Hint: The pnorm R function will be helpful.
Find the probability \(P(\overline{X} < 10)\). 💬
Find the probability \(P(4 \leq \overline{X} \leq 15)\). 💬
Find the probability \(P(\overline{X} > 13)\). 💬
🏡 Explain in words what each of the probabilities you have obtained in 1.2 above represent, in this context.
💻 Suppose that the probability that \(\overline{X}\) is less than a certain number of fuel tanks is equal to 40%. If the true population mean is \(\mu = 12\) fuel tanks per craft, and we are assessing our sample of \(n=25\) spacecraft, what must this number of fuel tanks be? 💬
Hint: You can use jamovi for this question too (think back to Computer Lab 4). Refer to the code chunk below for some guidance if you are stuck - you will have to fill in the missing ...’s:
# We can use the qnorm function for this type of question
qnorm(..., ..., ...)💻 Suppose that there is only a 10% probability of observing more than a certain number of fuel tanks for \(\overline{X}\) . If the true population mean is \(\mu = 12\), and we are assessing our sample of \(n=25\) spacecraft, what must this number of fuel tanks be? 💬
💻 Repeat 1.5, but this time suppose that there is only a 5% probability of observing more than a certain number of fuel tanks for \(\overline{X}\). 💬
💻 Suppose that, just as in 1.5 and 1.6, we are assessing the fuel tanks in our sample of \(n=25\) spacecraft, and we know that the true population mean is \(\mu = 12\) fuel tanks per craft. This time however, we would like to determine a range of values for the number of fuel tanks per craft, within which the majority of space craft will lie.
Firstly, we would like to determine the number of fuel tanks for \(\overline{X}\) such that there is only a 2.5% probability of observing less than this number of fuel tanks on a craft.
Secondly, we would like to determine the number of fuel tanks for \(\overline{X}\) such that there is only a 2.5% probability of observing more than this number of fuel tanks on a craft.
Compute these two values now, and report them as an interval (e.g. if the lower number is 8 and the higher number is 10, write the interval as (8, 10).) 💬
💻 Now that you have finished your inventory of this shipyard, you travel to another shipyard in a neighbouring state. Here, most of the craft are in use, so you are only able to assess \(20\) craft. You do not know the distribution of fuel tanks for the craft in this shipyard, but are told they have a mean of \(14\), and a standard deviation of \(7.55\).
Let \(\overline{X}\) denote the sample mean number of fuel tanks for craft from this new shipyard. Write down the distribution of \(\overline{X}\) in mathematical notation.
Suppose that you are engaged in an asteroid belt mining operation. \(60\) asteroids are currently being mined, and these represent a random sample taken from a population with an unknown distribution. However, scanning tools have revealed that the mean diameter of the asteroids in the belt is \(35\) kms, with \(\sigma = 32.5\).
💻 Let \(\overline{X}\) denote the sample mean. Write down the distribution of \(\overline{X}\) in mathematical notation. What special result does your answer rely upon? 💬
💻 Using this distribution, carry out the following calculations in jamovi. Remember, it can be helpful to sketch pictures of the distribution when working out these different probabilities.
For this question, for more accuracy, try entering the standard deviation as a fraction rather than a rounded number. For example, you hopefully found in 2.1 that \(\dfrac{\sigma^2}{n} = \dfrac{32.5^2}{60}\). Using R code, we can use ^ to represent powers (e.g. for squaring a number) and / for dividing, so that for the standard deviation we can use sd = sqrt(32.5^2 / 60).
Find the probability \(P(\overline{X} < 42)\). 💬
Find the probability \(P(18.5 \leq \overline{X} \leq 38)\). 💬
Find the probability \(P(\overline{X} > 26)\). 💬
\(P(\overline{X} < 30) + P(\overline{X} > 36)\) 💬
🏡 Explain in words what each of the probabilities you have obtained in 2.2 above represent, in this context.
🏡 A fellow miner claims that the last time they mined this asteroid belt, all the asteroids were at least 42 kms in diameter. Based on your sample data, do you believe they are telling the truth?
🏡 Suppose that another mining team, operating on a nearby asteroid cluster, has had to haul \(15\) asteroids back to your station for further processing. The distribution of the asteroid cluster is unknown, with \(\mu = 19.25\) and \(\sigma^2 = 8.1\). Let \(\overline{X}\) denote the sample mean. What is the distribution of \(\overline{X}\)? Make sure to clearly explain your answer.
🏡 To further consolidate your understanding of the central limit theorem, check out the interactive web application in the Topic 4 readings.
If you have any questions, discuss with your classmates and/or your computer lab demonstrator.
These notes have been prepared by Amanda Shaker and Rupert Kuveke. The copyright for the material in these notes resides with the authors named above, with the Department of Mathematical and Physical Sciences and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License BY-NC-ND.