Stats quiz 2

When it is not possible to calculate the Population Mean (μ) and the Population Standard Deviation (σ), we need to use the Point Estimators calculated from sample data (xand s). These do not exactly match the Population Parameters (μand σ) but they provide good estimates. Using these estimates based on sample data to approximate the population parameters is what Inferential Statistics is all about. The next thing we would like to know is how good are these estimates. As you can see from the example we are doing, the estimates are close to the real values, but they are not exact. Is there any way we can see how close they are likely to be? With the help of the Central Limit Theorem, the answer is Yes. Remember that every time we take a different sample, using a different Line in the Random Number Table, we are going to get a different value for x, the Sample Mean. This means that x itself is a Random Variable. According to the CLT, xwill have approximately a Normal Distribution with a mean μ x = μand a standard deviation, . This standard deviation of xitself is called the Standard Error of the Mean. We denote it by the symbol instead of just σ, to distinguish the Standard Error of the Mean from the Standard Deviation of the original variable x. The formula for calculating the Standard Error of the Mean is given below: σ x = σ n Use this formula to calculate the Standard Error of the Mean for this example, where our sample size n = 5. Give two decimal places in your answer.

2.41

Now, use the index numbers from the previous question to identify the values of the variable x (Number of Cousins) that you will use for your statistical calculations.

5 2 9 6 23

Calculate the actual Population Standard Deviation using all of the data for the entire population. This is possible to do on a calculator if you are careful, but you might find it easier to use a computer. Make sure you use the formula for σ, the population standard deviation. Remember that this formula is slightly different than the formula used for s, the sample standard deviation. Use two decimal places in all of your calculations and round your final answer off to two decimal places.

5.39

In this example, because we have all of the population data, it will be possible to calculate the exact population standard deviation, σ. But normally you do not have any way of knowing the exact population standard deviation because you do not have all of the population data. You have to estimate it using the sample data. Assume that you only took one sample, the one we just did using Line 6 in the Random Number Table. Based on this sample only, if that was all the information you had, what would be your best guess of the value of the population standard deviation, σ? This is called the Point Estimator of the Population Standard Deviation. Give two decimal places in your answer.

8.22

Use the values of x from your sample, along with the sample mean, to calculate the sample standard deviation, s. Remember to use the formula for the sample standard deviation, which is different from the formula for the population standard deviation. Give two decimal places in your answer.

8.22

Now, use all of the data from the entire population and calculate the population mean. Use the formula we learned in Module 4 for μ. You might find it easier to use a computer to calculate this, but it can also be done on a calculator. Give one decimal point in your answer.

8.5

In this example, because we have all of the population data, it will be possible to calculate the actual population mean, μ, and we will do that a few questions later. But normally you do not have any way of knowing the population mean because you do not have all of the population data. You have to estimate it using sample data. Assume that you only took one sample, the one we just did using Line 6 in the Random Number Table. Based on this sample only, if that was all the information you had, what would be your best guess of the value of the population mean, μ? This is called the Point Estimator of the Population Mean.

9

Use the values of x from the previous question to calculate the value of the sample mean (x) for your sample. Remember to use the x values, and not the index numbers, in your calculation.

9

Did your Point Estimator of the Population Mean (based on your sample) match the actual Population Mean? The answer is not surprising because you did not use all of the data from the entire population when you calculated the sample mean.

no

Did your Point Estimator of the Population Standard Deviation (calculated a few questions earlier) match the actual Population Standard Deviation? The answer should not be surprising because the Point Estimator was based on a sample, and did not include all of the data used to calculate the actual Population Standard Deviation.

no

According to the CLT, xhas approximately a Normal Distribution. That means that we can calculate z-scores and calculate probabilities that xwill have certain values using the z-scores and the Standard Normal Distribution probability table. Calculate the z-score for your sample value of x. The formula you use for this is: z x = x − μ x σ x Remember that since this is a z-score for xyou need to use μ xand σ xin your formula for z. Refer to previous questions to get these values. Z-scores are almost always reported with two decimal places, so round your answer to two decimals places.

.21

What is the difference between your sample mean and the population mean? This is called the Error of Estimation and it is calculated using the formula x − μ. (Normally, it will not be able to calculate the value of the Error of Estimation because normally we only know xand we don't know μ. But in this example we know both of them so we can calculate it.)

.5

Use the z-score you calculated in the last question to find the probability of getting a z-score equal to that value or higher. Look up the z-score in the Standard Normal Table, but remember that since this is a greater than probability you will need to use the one-minus rule to find the correct answer. Since the table shows probabilities with four decimal places, give four decimal places in your answer.

0.4168

Be sure to read the instructions and look at the example in the Module 10.90 document before beginning this assignment. You will also need to refer to the data table and random number table in that document. Let's pick another random sample from the same population. This time, instead of using Line 48, we will use Line 6. Follow the same process and indicate the five index numbers of the observations you will include in your sample based on Line 6.

10 7 11 18 6