In a particular year, the percentage rates of return of U.S....

In a particular year, the percentage rates of return of U.S. common stock mutual funds had a normal distribution with a mean of $14.8$ and a standard deviation of 6.3. A random sample of nine of these mutual funds was taken. a. What is the probability that the sample mean percentage rate of return is more than $19.0$ ? b. What is the probability that the sample mean percentage rate of return is between $10.6$ and $19.0$ ? c. The probability is $0.25$ that the sample mean percentage return is less than what number? d. The probability is $0.10$ that the sample standard deviation of percentage return is more than what number? e. If a sample of 20 of these funds was taken, state whether the probability of a sample mean percentage rate of return of more than $19.0$ would be smaller than, larger than, or the same as the correct answer to part (a). Sketch a graph to illustrate your reasoning.

In a particular year, the percentage rates of return of U.S. common stock mutual funds had a normal distribution with a mean of $14.8$ and a standard deviation of 6.3. A random sample of nine of these mutual funds was taken.
a. What is the probability that the sample mean percentage rate of return is more than $19.0$ ?
b. What is the probability that the sample mean percentage rate of return is between $10.6$ and $19.0$ ?
c. The probability is $0.25$ that the sample mean percentage return is less than what number?
d. The probability is $0.10$ that the sample standard deviation of percentage return is more than what number?
e. If a sample of 20 of these funds was taken, state whether the probability of a sample mean percentage rate of return of more than $19.0$ would be smaller than, larger than, or the same as the correct answer to part (a). Sketch a graph to illustrate your reasoning.

Key Concepts

Effect of Sample Size on Variability

The size of the sample has a direct impact on the variability of the sample mean. As the sample size increases, the standard error decreases, meaning that the distribution of the sample mean becomes more concentrated around the true population mean. This concept is crucial when comparing probabilities of extreme sample means between different sample sizes.

Chi-Square Distribution

When samples are taken from a normally distributed population, the sample variance (and by extension the sample standard deviation) has a distribution that is related to the chi-square distribution. This relationship, involving degrees of freedom, allows for the determination of probabilities and critical values related to the variability of the sample data.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of sample means obtained by drawing multiple samples of a fixed size from a population. When the population is normally distributed, the sample mean is also normally distributed, regardless of the sample size, with the same mean as the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, with a bell-shaped curve. It is fully characterized by its mean and standard deviation, making it ubiquitous in statistics as it often models real-world phenomena where data tend to cluster around a central value with a known spread.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It quantifies the amount of variation or dispersion of the sample means around the population mean, and it decreases as the sample size increases, reflecting more precise estimates of the population parameter.

Transcript

00:01 Hello, so here in part a, we're interested in computing the probability that the sample mean percentage of return is higher than 19.

00:08 So we want to compute the probability that x bar is greater than 19.

00:12 So using the central limit theorem we have here that probability is going to be equal to the probability that z is greater than 19 minus 14 .8, all divided by 2 .1, which is then going to be equal to 1 minus the probability that z is less than or even.

00:30 Equal to 2, so that's going to be 1 minus 0 .972, which is going to be equal to 0 .0 -228.

00:46 And then we have that the sample mean percentage rate is higher than 19 % is going to be 0 .0228.

00:54 And then in part b, we want to compute the probability that 10 .3.

01:03 6 is going to be less than equal to x bar, which is going to be less than equal to 19.

01:10 So again, using the central limit theorem, this is going to be equal to the probability that z is less than or equal to 19 minus 14 .8 all over 2 .1.

01:23 It gives us the probability that z is less than or equal to 2, minus the probability that z is less than or equal to 10 .6 minus 14 .8 all over 2 .1, giving us the probability here that z is less than or equal to negative 2.

01:40 That's going to be 0 .972 minus 0 .0 -225, which is going to be equal to 0 .9545.

01:52 So that's going to be the probability that the sample mean percentage rate of returns is between 10 .6 and 19 is going to be 0 .9545.

02:08 And then in part c, we want to find the value below which 25 % of the sample means of rate of returns lie.

02:17 So we're finding here the value of little x bar such that the probability of x bar, is less than or equal to little x bar, which is going to be equal to 0 .25.

02:33 So here, from our normal table, we get the probability that z is less than or equal to negative 0 .6745 is going to be equal to 0 .25.

02:46 So we then have x bar minus 14 .8, all over 2 .1, is equal to negative 0 .6745, giving us that x bar is equal to negative 0 .6745 times 2 .1 plus 14 .8 is going to give us that the value below which 25 % of the sample of returns lie is going to be 13 .3836%.

03:25 And then in part d, we consider the, get the sample standard deviation...

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