Question

In: Statistics and Probability

Let's try putting the CLT (Central Limit Theorem) to practice. In a study of bumblebee bats,...

Let's try putting the CLT (Central Limit Theorem) to practice. In a study of bumblebee bats, one of the world’s smallest mammals, the weights have a mean of 2.0 grams and a standard deviation of 0.25 gram.

a) Find the probability that a randomly selected bat from the study weights more than 2.3 grams.

b) A random sample of 7 bumblebee bats is selected from the study. Find the probability that the mean weight of the sample is more than 2.3 grams.

c) A random sample of 40 bumblebee bats is selected from the study. Find the probability that the mean weight of the sample is more than 2.3 grams.

Expert Solution

Part a)

P ( X > 2.3 ) = 1 - P ( X < 2.3 )
Standardizing the value

Z = ( 2.3 - 2 ) / 0.25
Z = 1.2

P ( Z > 1.2 )
P ( X > 2.3 ) = 1 - P ( Z < 1.2 )
P ( X > 2.3 ) = 1 - 0.8849
P ( X > 2.3 ) = 0.1151

Part b)

P ( X > 2.3 ) = 1 - P ( X < 2.3 )
Standardizing the value

Z = 3.17

P ( Z > 3.17 )
P ( X > 2.3 ) = 1 - P ( Z < 3.17 )
P ( X > 2.3 ) = 1 - 0.9993
P ( X > 2.3 ) = 0.0007

Part c)

P ( X > 2.3 ) = 1 - P ( X < 2.3 )
Standardizing the value

Z = 7.59

P ( Z > 7.59 )
P ( X > 2.3 ) = 1 - P ( Z < 7.59 )
P ( X > 2.3 ) = 1 - 1
P ( X > 2.3 ) = 0

orchestra answered 1 year ago

Central Limit Theorem (CLT) . Investigate the parameters CLT could be applicable to? Explain in detail...

Central Limit Theorem (CLT) . Investigate the parameters CLT could be applicable to? Explain in detail by writing an essay without plagiarism. Min 1.5 page Max 2 pages

A) state the complete Central Limit Theorem (CLT) B) explain why we need the theoretical idea...

A) state the complete Central Limit Theorem (CLT) B) explain why we need the theoretical idea of sampling distributions in a hypothesis test even though we only take one sample to decide between the hypothesis. C) relate each part of the formula r= X-Mean0 / S/ square root of n D) Explain what type 1 and type 2 errors are E) explain how it is possible to conduct the correct test flawlessly using a simple random sample of sufficient size...

This week we’ve introduced the central limit theorem. According to the central limit theorem, for all...

This week we’ve introduced the central limit theorem. According to the central limit theorem, for all samples of the same size n with n>30, the sampling distribution of x can be approximated by a normal distribution. In your initial post use your own words to explain what this theorem means. Then provide a quick example to explain how this theorem might apply in real life. At last, please share with us your thoughts about why this theorem is important.

1. What is the Central Limit Theorem? Try to state it in your own words. 2....

1. What is the Central Limit Theorem? Try to state it in your own words. 2. Consider the random variable x, where x is the number of dots after rolling a die. Make a sketch of the probability distribution of this variable. What is the expected value of x? 3. Now consider the random variable that is the average number of dots after four rolls. Is this variable normally distributed? Explain. 4. Suppose we changed the definition to the average...

To see for yourself how the central limit theorem works, let's say we have a normal...

To see for yourself how the central limit theorem works, let's say we have a normal distribution (with mean =100 and standard devation = 20). Let's generate some random samples of various sizes from this distribution. We can do this in excel using =norm.inv(rand(),100,20) and it will randomly generate numbers from this distribution. I generated four samples of size 5, 10, 20 and 30, and got the means of 124 (n=5); 91 (n=10); 105 (n=20); 103 (n=30). If I continue...

It is said that the Central Limit Theorem is the most important theorem in all of...

It is said that the Central Limit Theorem is the most important theorem in all of Statistics. In your own words, describe why it is so important.

To better understand how the Law of Large Number (LLN) and the Central Limit Theory (CLT)...

To better understand how the Law of Large Number (LLN) and the Central Limit Theory (CLT) works, we can use the following simulation to demonstrate them. (Note: attach the computer codes which implement the analysis in your answer) To demonstrate the Lindberg-Levy CLT, please generate 1000 sets of independent random samples, each has 1000 observations, from i.i.d X^2(5) distribution. First choose one set of the random samples and plot a histogram. Do you see a X^2(5) distribution? Denote the sample...

describe what the central limit theorem is, and list the requirements necessary for using the Central...

describe what the central limit theorem is, and list the requirements necessary for using the Central Limit Theorem.

3.27. Problem. (Section 11.5) The following are applications of Theorem 11.6 or the Central Limit Theorem....

3.27. Problem. (Section 11.5) The following are applications of Theorem 11.6 or the Central Limit Theorem. (a) Determine the distribution of (1/5)X1 + (2 /5)X2 + (2/5)X3 if X1, X2 and X3 are independent normal distributions with µ = 2 and σ = 3. (b) The weight (kg) of a StarBrite watermelon harvested under certain environmental conditions is normally distributed with a mean of 8.0 with standard deviation of 1.9. Suppose 24 StarBrite watermelons grown in these conditions are harvested;...

What is the central limit theorem? What is the standard error of the mean?

Question 5 What is the central limit theorem? Provide examples. Question 6 What is the standard error of the mean? Provide examples.