Small Sample Weights of M&M plain candies are normally dis...

Question

Small Sample Weights of M&M plain candies are normally distributed. Twelve M&M plain candies are randomly selected and weighed, and then the mean of this sample is calculated. Is it correct to conclude that the resulting sample mean cannot be considered to be a value from a normally distributed population because the sample size of 12 is too small? Explain.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A group of researchers are studying the distribution of heights of sunflower plants located in a large field. They randomly select 15 sunflower plants and measure their heights to calculate the mean height of their sample. Is it accurate to state that the sample mean height cannot be considered as coming from a normally distributed population because the sample size is less than 30? Awesome. So it appears for this particular problem, we're asked to determine whether or not it's considered accurate to state that this particular sample mean height cannot be considered as coming from a normally distributed population, because the sample size is less than 30. So we're trying to take this statement, and we're trying to figure out its accuracy. So we're trying to come up with an explanation. To whether or not this statement is accurate. So now that we know what we're trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer might be. A is yes, because the sample size is too small. B is no, because the sample mean will be normally distributed if the population is normally distributed, regardless of sample size, and C is inconclusive without more data. And D is no, because a distribution is only normal when the central limit theorem applies. Awesome. So, With that in mind, our first step. we need to recall and make sure we understand what the central limit theorem is, which we'll call this CLT for short. And as we should recall, the central limit theorem suggests that the distribution sample means will approximate a normal distribution as the sample size increases. But it does not strictly prohibit smaller sample sizes from being normally distributed, especially if the population distribution is normal. Our second step is we need to consider our sample size. So considering our sample size, as we should note and recall, while larger sample sizes are generally preferred for the central limit theorem to hold more accurately, a sample size of 15 does not automatically disqualify. The sample mean from from being considered as coming from a normally distributed population, so size doesn't matter. Our third step is we need to consider the population distribution. So as we should note and recall, if the population of sunflower plant heights are normally distributed, then the sample mean of 15 sunflower plants will also be normally distributed, regardless of the sample size being less than 30. And our 4th and final step is we need to make our conclusions. So with everything we have gone over, it is incorrect. It's going to be incorrect to conclude that the sample mean height cannot be considered to come from a normally distributed population based on the sample size of 15. The key factor here that we need to focus our attention on, as we should recall, is the distribution of the population, not just the sample size. And because of that, our final answer, without a doubt has to be multiple choice answer letter B. Which is no, because the sample mean will be normally distributed if the population is normally distributed, regardless of sample size. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Central Limit Theorem

Normal Distribution

Sample Size and Statistical Inference

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