During a recent period of one year, the mean percent increase ...

Question

During a recent period of one year, the mean percent increase in value on Wednesdays of the cryptocurrency Dogecoin was 7.46%, with a standard deviation of 53.47%. Random samples of size 50 are drawn from this population and the mean of each sample is determined. (Source: Crypto Indicators)

c. What is the probability that the mean percent increase for a given sample is between −10% and 30%?

c. What is the probability that the mean percent increase for a given sample is between −10% and 30%?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a certain stock's daily percent return on Fridays as a mean of 3.12% and a standard deviation of 41.25%. If random samples of 40 days are selected and they mean return for each sample is calculated, what is the probability that a sample mean is between -5% and 10%? And we're given four possible choices as our answers. For choice A, we have 0.5395. For choice B, we have 0.5517. For choice C, we have 0.6823. And for choice D, we have 0.7476. Now, we need to find the probability that a sample mean is between -5% and 10%. And we're told that we have a random sample of 40 days. That means our sample size N is going to be equal to 40. Now, according to the central limit theorem, if random samples of size n, where n is greater than or equal to 30 are drawn from any population with a mean u and a standard deviation sigma, then the sampling distribution of sample means approximates a normal distribution. Now our case here we have N equal to 40, which is actually greater than 30. That means that the sampling distribution of sample means approximates a normal distribution. So that means that our mean of sample means, which we'll label as mu, the X bar, is just going to be equal to our population mean mu, which in this case is going to be equal to the 3.12%, which will convert into a decimal by divided by 100, so this will give us 0.0312. Also, our standard deviation of sample means, which is gonna be sigma of X bar, we call that that's just gonna be equal to our population standard deviation sigma, divided by the square root of N. Now in our case here, our standard population standard deviation is the 41.25%. So when we convert that into a decimal and substitute it into our formula here, you get 0.4125, and that's going to be divided by the square root of 40. And when we evaluate this expression, this is going to be approximately equal to. 0.0652. Now, since we're looking for a sample, meaning that's between -5% and 10%, we want to convert those values into a Z score. So we call your formula for finding the Z score, that's gonna be Z. is equal to the quantity of X minus UX bar in quantity divided by sigma sub x bar. So we're going to have two Z values that we need to calculate. So for the first Z value, we'll call it Z1. This is going to be correspond to the -5%, so that means that Z1 is going to be equal to the quantity of minus 0.05 minus. 0.0312 in quantity divided by 0.0652. So here we've substituted in our values for use of X bar and Sigma sub X bar. And when we evaluate this expression, this is going to be approximately equal to. -1.245. Now, we'll have to calculate the Z score for the 10%, so we'll label that as Z2, and this is going to be equal to the quantity of 0.10 minus 0.0312 in quantity, divided by 0.0652. And when we evaluate this expression, this is going to be approximately equal to. 1.055. So now we want to find the cumulative probabilities corresponding to these two Z scores using our standard normal distribution table. And so if we look at Z1, when we look this up in our table, we see that it falls between two of the values, Z values in our table. It falls between Z equal to minus 1.24. Which has a cumulative probability of 0.1075, and the other Z value is going to be Z equal to minus 1.25, which has a cumulative probability of. 0.1056. Now, since our Z-value falls between two of the Z values on our table, to find its cumulative probability, we'll need to use interpolation. So, our cumulative probability that we're looking for, which is P of Z less than minus 1.245, is going to be equal to, and here we call your formula for interpolation. So first, we'll have our change in our probabilities divided by our change in Z scores from our table. So we'll have the quantity. Of 0. 1075 minus 0.1056, and that quantity is going to be divided by the quantity of minus 1.24, minus 1.25 in quantity. And then we'll have to multiply this fraction by the quantity of our Z score that we're looking for, which is -1.245. Minus the smaller of the two Z values from our table, which in this case is going to be -1.25. In quantity, then we'll have to add to this the cumulative probability for Z equal to -1.25, and that's going to be 0.1056. And so when we evaluate this expression, we're going to get that this cumulative probability is approximately equal to. 0.1066. Now we have to do something similar for Z2. So again, if we look up our value for Z2 in our table, we see that it falls between two of the entries in our table. It falls between Z equal to 1.05, which has a cumulative probability of. 0.8531, and the other value is going to be Z equal to 1.06, which has cumulative probability of. 0.8554. So, again, we'll have to use interpolation to find our cumulative probability for Z2. So we're looking for the cumulative probability P of Z less than 1.055. Which is going to be equal to, and again, we'll need to use our trippolation formula, so we'll have the quantity or change in our probabilities. So we'll have 0.8554 minus 0.8531 in quantity, divided by. The change in our Z values, which is the quantity of 1.06 minus 1.05 in quantity, then that gets multiplied by the quantity of 1.055 minus 1.05 in quantity, then we'll have plus. The 0.8531. And when we evaluate this expression, this is going to be approximately equal to. 0.85. 42. So we'll now use these cumulative probabilities to calculate the probability that we're asking to find in the problem. So that probability. Which is the probability P. Of minus 0.05 is less than X bar, which is less than 0.10. That means that our sample mean is between -5% and 10%. So, for that probability, That's going to be equal to. The probability P of minus 1.245 is less than Z, which is less than 1.055. Since those are the Z values that correspond to the -5% and 10%. However, we can write that probability. Uh, in terms of Z as being equal to P of Z less than 1.05 5. Minus P of Z, less than -1.245. And we actually have values for both those cumulative probabilities. That means that this is going to be equal to. 0.8542. Minus 0.1066. And when we evaluate this expression, this is going to be equal to. 0.7476. So this here is the answer that we were looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Central Limit Theorem

Standard Error

Z-scores

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