Hello I need help answering these questions .

1Read Demonstration (Focus on Problem Solving) on page 272; and Demonstration 8.1 on page 273; and explain the four major steps involved in Hypothesis Testing. Detail which step you feel will be the most important to the process and why.

2) Give a real life example of how you might use hypothesis testing.

3) What are the easiest and hardest concepts in the chapter? Explain your answer

The Shape of the t Distribution The exact shape of a t distribution changes with degrees of freedom. In fact, statisticians speak of a "family" of t distributions. That is, there is a different sampling distribution of t (a distribution of all possible sample t values) for each possible number of degrees of freedom. As df gets very large, the t distribution gets closer in shape to a normal z-score distribution. A quick glance at Figure 9.1 reveals that distributions of t are bell-shaped and symmetrical and have a mean of zero. However, the t distribution has more variability than a normal z distribution, especially when df values are small (see Figure 9.1). The t distribution tends to be flatter and more spread out, whereas the normal z distribution has more of a central peak.

The reason that the t distribution is flatter and more variable than the normal z-score distribution becomes clear if you look at the structure of the formulas for z and t. For both z and t, the top of the formula, M- , can take on different values because the sample mean (M) varies from one sample to another. For z-scores, however, the bottom of the formula does not vary, provided that all of the samples are the same size and are selected from the same population. Specifically, all the z-scores have the same standard error in the denominator, M=2/n , because the population variance and the sample size are the same for every sample. For t statistics, on the other hand, the bottom of the formula varies from one sample to another. Specifically, the sample variance (s2) changes from one sample to the next, so the estimated standard error also varies, sM=s2/n . Thus, only the numerator of the z-score formula varies, but both the numerator and the denominator of the t statistic vary. As a result, t statistics are more variable than are z-scores, and the t distribution is flatter and more spread out. As sample size and df increase, however, the variability in the t distribution decreases, and it more closely resembles a normal distribution.

Determining Proportions and Probabilities for t Distributions Just as we used the unit normal table to locate proportions associated with z-scores, we use a t distribution table to find proportions for t statistics. The complete t distribution table is presented in Appendix B, and a portion of this table is reproduced in Table 9.1. The two rows at the top of the table show proportions of the t distribution contained in either one or two tails, depending on which row is used. The first column of the table lists degrees of freedom for the t statistic. Finally, the numbers in the body of the table are the t values that mark the boundary between the tails and the rest of the t distribution.

Table 9.1. A portion of the t-distribution table. The numbers in the table are the values of t that separate the tail from the main body of the distribution. Proportions for one or two tails are listed at the top of the table, and df values for t are listed in the first column.

Proportion in One Tail

0.25

0.10

0.05

0.025

0.01

0.005

Proportion in Two Tails Combined

df

0.50

0.20

0.10

0.05

0.02

0.01

1

1.000

3.078

6.314

12.706

31.821

63.657

2

0.816

1.886

2.920

4.303

6.965

9.925

3

0.765

1.638

2.353

3.182

4.541

5.841

4

0.741

1.533

2.132

2.776

3.747

4.604

5

0.727

1.476

2.015

2.571

3.365

4.032

6

0.718

1.440

1.943

2.447

3.143

3.707

For example, with df=3 , exactly 5% of the t distribution is located in the tail beyond t=2.353 (Figure 9.2). The process of finding this value is highlighted in Table 9.1. Begin by locating df=3 in the first column of the table. Then locate a proportion of 0.05 (5%) in the one-tail proportion row. When you line up these two values in the table, you should find t=2.353 . Because the distribution is symmetrical, 5% of the t distribution is also located in the tail beyond t=-2.353 (see Figure 9.2). Finally, notice that a total of 10% (or 0.10) is contained in the two tails beyond t=2.353 (check the proportion value in the "two-tails combined" row at the top of the table).

Details The t distribution with df=3 . Note that 5% of the distribution is located in the tail beyond t=2.353 . Also, 5% is in the tail beyond t=-2.353 . Thus, a total proportion of 10% (0.10) is in the two tails combined.

A close inspection of the t distribution table in Appendix B will demonstrate a point we made earlier: as the value for df increases, the t distribution becomes more similar to a normal distribution. For example, examine the column containing t values for a 0.05 proportion in two tails. You will find that when df=1 , the t values that separate the extreme 5% (0.05) from the rest of the distribution are t=12.706 . As you read down the column, however, you should find that the critical t values become smaller and smaller, ultimately reaching 1.96. You should recognize 1.96 as the z-score values that separate the extreme 5% in a normal distribution. Thus, as df increases, the proportions in a t distribution become more like the proportions in a normal distribution. When the sample size (and degrees of freedom) is sufficiently large, the difference between a t distribution and the normal distribution becomes negligible.

Caution: The t distribution table printed in this book has been abridged and does not include entries for every possible df value. For example, the table lists t values for df=40 and for df=60 , but does not list any entries for df values between 40 and 60. Occasionally, you will encounter a situation in which your t statistic has a df value that is not listed in the table. In these situations, you should look up the critical t for both of the surrounding df values listed and then use the larger value for t. If, for example, you have df=53 (not listed), look up the critical t value for both df=40 and df=60 and then use the larger t value. If your sample t statistic is greater than the larger value listed, you can be certain that the data are in the critical region, and you can confidently reject the null hypothesis.