Refer to Exercise 7.9 . Assume now that the amount of fill dispensed by the bottling machine is

Refer to Exercise 7.9 . Assume now that the amount of fill...

Key Concepts

Impact of Population Standard Deviation

The population standard deviation (?) directly influences the spread of both the population and the sampling distributions. When comparing scenarios with different ? values, such as ? = 2 versus ? = 1, the probability calculations and margins of error differ, reflecting how the intrinsic variability of the population affects the confidence in sample estimates.

Effect of Sample Size on Precision

As the sample size increases, the standard error decreases, which in turn increases the likelihood that the sample mean will lie closer to the population mean. This concept is highlighted by comparing probabilities for different sample sizes, showing that larger samples yield more precise estimates of the population mean.

Margin of Error and Probability Calculations

In this context, the margin of error specifies a range around the population mean (?) within which the sample mean is likely to fall. Calculating the probability that the sample mean differs from ? by no more than a fixed amount involves using the standard normal distribution, particularly the z-score transformation applied to the standard error.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of the sample mean computed from a population. When samples are drawn from a normally distributed population, the sample means will also be normally distributed, regardless of the sample size, with the same mean (?) as the population but a reduced spread, as measured by the standard error.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell shape, defined by its mean (?) and standard deviation (?). It is important because many phenomena naturally follow this pattern, and it forms the basis for many statistical methods and probability calculations, including those involving measurements, sampling distributions, and hypothesis testing.

Standard Error

The standard error quantifies the dispersion of the sampling distribution of the sample mean and is calculated by dividing the population standard deviation (?) by the square root of the sample size (n). This measure is crucial for understanding how the variability of the sample mean decreases with increasing sample size, thereby affecting the precision of estimates.

Transcript

00:01 With this problem, you're basically redoing problem 9 with a different standard deviation.

00:09 So in problem 7 .9, we had a standard deviation of 1.

00:15 And in this particular problem, we're going to do the problem with a standard deviation of 2.

00:23 And the problem is we want to find the probability that the sample mean is within, 0 .3 ounces of the true population mean with various sample sizes.

00:40 So for part a, we're going to do the sample size being 9.

00:45 In part b, we're going to change the sample size to be 25, 36, 49, and 64.

00:55 So let's start with the absolute value inequality.

01:04 And we are going to to convert that to a compound inequality.

01:22 And now i want you to recall the z score formula, and the z score formula is x bar minus mu divided by sigma over the square root of n.

01:33 And in our compound inequality, we have the numerator of that z score.

01:40 So if i were to divide that center part of the compound inequality by the expression sigma divided by the square root of n, then that center can be converted to a z.

01:55 But i can't just do that without dividing all the parts by that expression.

02:05 So we now have a new probability statement, negative 0 .3 over sigma divided by square root of n is less than or equal to z, which is less than or equal to 0 .3 over sigma divided by the square root of n.

02:20 Now we know for all the parts in this problem, we are now going to let the population standard deviation to be two.

02:30 So i'm going to rewrite that as the probability of negative 0 .3 over 2 divided by the square root of n is less than or equal to z, which is less than or equal to 0 .3 over 2 divided by the square root of n.

02:48 Now by doing that, we have a now created compound fractions.

02:52 We've got a fraction within a fraction.

02:55 So if i were to multiply the top and the bottom by a square root of n in both parts here, what will happen in the denominator is the square root of n will cancel with the square root of n.

03:13 So now we've got a new probability statement.

03:16 So we have the probability of negative 0 .3 times the square root of n over 2 is less.

03:24 Less than or equal to z, which is less than or equal to positive point three times the square root of n over two.

03:34 And now that'll be relatively easy to use with the various sample sizes.

03:41 So let's do part a.

03:43 Part a, we wanted if n equals 9.

03:48 Well, if n equals 9, then we are working on the probability of negative 0 .3 times the square root of 9.

03:56 Over 2 is less than or equal to z, which is less than or equal to 0 .3 times the square root of 9 over 2.

04:05 And since the square root of 9 is 3, we can replace each of these with just a plain old 3.

04:14 So now we would end up with a 0 .3 times a 3, or we could say negative 0 .9 over 2 .2.

04:28 Is less than or equal to z, which is less than or equal to positive 0 .9 over 2.

04:34 And if we were to divide 0 .9 by 2, we would have the statement, the probability of negative 0 .45 is less than or equal to z, which is less than or equal to positive 0 .45.

04:52 So if you think about what that picture would look like, we've got a bell -shaped curve and we would have negative 0 .45 on the left and positive 0 .45 on the right and we're trying to find the probability that we're in between.

05:09 Now unfortunately our normal distribution table, table 4 in the back of your textbook, only talks about the area in the right tail.

05:20 So we are concerned with the z score being greater than 0 .45.

05:30 And if we were to find the area in the right tail, which will also match the area of the left tail, we can then talk about two times that probability.

05:42 And the fact that the entire curve is one, if we take one, take away those two tails, we will be left with the orange shaded region that we were after.

05:56 So you will need to go to your table in the back of your textbook.

06:00 Again, it's table four.

06:02 And you will find z in the left -hand column.

06:07 And when you look down that column, it only goes out to the tenth decimal place.

06:14 So we're going to look up 0 .4.

06:17 And across the top, we're going to find the second decimal place.

06:22 You're going to look under the 0 .5.

06:24 And in doing so, you're going to look at the 0 .4.

06:27 Going to get a value of 0 .3 -264, and then if i do 1 minus twice that value, i'm going to get a probability of 0 .3472 of my sample mean being within 0 .3 ounces of my population mean when my sample size is 9.

06:57 I'm going to utilize that same formula.

07:01 Again, we're utilizing this formula, and we're just going to change the sample size for part b.

07:07 So in part b, we are now going to let n equal 25.

07:15 So we will have the probability of negative 0 .3 times the square root of 25 over 2 is less than or equal to z, which is less than or equal to 0 .3 times the square root of 25.

07:29 Over 2.

07:30 We can simplify the square root of 25 into a 5.

07:36 So we can turn this into a 5 and this into a 5.

07:41 So in essence we're multiplying negative 0 .3 times a 0 .5.

07:46 And by doing that, we get negative 1 .5 over 2 is less than or equal to z, which is less than or equal to positive 1 .2.

07:59 5 over 2 and negative 1 .5 divided by 2 is the same thing as negative 0 .75.

08:10 So we are now looking for the probability that your z score is in between negative 0 .75 and positive 0 .75.

08:20 So again, you're going to go to your table...