Need help in solving these questions!

Question

How do you solve these problems in Statistics? The topic is about Testing the Difference between two population means case of two independent samples.

I really am finding it hard to solve these questions. It's only 2 questions though and it's base on testing hypothesis, so it has steps on how to solve it. But this time it has 2 population means unlike the other one is only one population mean. But the problem is, how to do you solve these 2 questions? If it's okay, can you provide a detailed steps and how you got it? I seriously spent a day on thinking how to get the answers but to no avail. Please help me Wyzanters! :(

 

Here are the questions:

1. Do you view chocolate bar as delicious or fattening? Your attitude may depend on your gender. A researcher examined the importance of food as a source of pleasure versus concerns about food associated w/ weight gain and health. The following results were obtained;

 

Males                                                     Females

n= 31                                                    n= 32

Sample mean= 29                                  Sample mean= 35

Sample SD= 3.4                                     Sample SD= 5.6

 

Based on the results, is there a significant difference between the attitudes of males and females? Level of significance is 0.01

 

 

2. A course in Psych is taught to 10 students by the traditional method.  Another group of 11 students was given the same course by means of another method.  At the end of the semester, the same test was administered to each group.  The 10 students under method A made an average of 82 with a standard deviation of 5, while the eleven students under method B made an average of 78 with a standard deviation of 6.  is there a significant difference in the performance of the two groups of students at 5% level of significance? 

Answer 1

Paige:

 

Question 1.

 

You MUST assume the following :- (1) The original data are normally distributed with same variance (or Std Dev).  (2) Each of the sample is a random sample. (3) The two samples are independently obtained.  If any of the assumptions is not true, the conclusions MAY NOT BE VALID.

 

Step 1. Calculate the unbiased estimate of the common variance, and then the standard error.

The sample variance of Sample 1  (called S₁²) is (given Sample SD)²  = (3.4)² = 11.56

The sample variance of Sample 2  (called S₂²) is (given Sample SD)²  = (5.6)² = 31.36

 

(RED FLAG, the sample estimates of individual variances are VERY different, so the assumption that the population variances are the SAME may not be TRUE!)

 

Compute the estimate of the common variance = [(n₁ - 1) S₁² + (n₂ -1) S₂²] / (n₁ + n₂ - 2) = [(31-1)(11.56) + (32 -1)(31.36)] / (31+32-2) = 21.6223.

Std Error (Say SE) is SE = sqrt (21.6223) = 4.65.

 

Step 2. Compute the test statistic.  t = [x₁ bar - x₂ (bar)] / {(SE) sqrt [(1 / n₁) + (1 / n₂)]}

t = [29 - 35] /{4.65 [sqrt (1/31 + 1/32)]} = - 5.12 (if my arithmetic is correct)

 

Now set up the null and the alternative hypotheses.

 

Step 3.  Null hyp: (mu)₁ = (mu)₂ , where (mu)₁ and (mu)₂ are the respective population means.

Since the question asks " is there a significant difference......" and the difference is not in one direction or the other, the alt. hyp is (mu)₁ NOT EQUAL TO (mu)₂.

The alternative hypothesis is two-sided. So a two sided rejection is used. If the absolute value of the computed t-statistic is greater than an an appropriately chosen "critical value" (see below) for the given level of significance, then the data provide sufficient evidence against the null hypothesis at the given significance level, that is the "difference is significant" at the given significance level.

 

Step 4.  Obtain the critical value.

 

Since both sample sizes are "large" (rule of thumb, each greater than 30), the critical value is obtained from the standard normal tables.  The given sig. level is 0.01.  Since the rejection region  is two sided, the sig level is divided in half. 0.01/2 = 0.005.  From the standard normal tables the "z_value" corresponding to 0.005 (such that the area under the standard normal curve to the right of the z_value is 0.005 is obtained.  It comes out to be 2.576.

 

Step 5.  Do the test.

 

Since the absolute value of the calculated t_statistic (Step 2) is |- 5.12| = 5.12 AND it is larger than 2.576, we say that the data provide sufficient evidence to reject the null hypothesis that the population means of females and males (for WHATEVER was measured, the problem does NOT state that) are the same and that there is a significant difference between the attitudes of males and females at 0.01 level of significance.

 

Paige, please follow and understand all the steps.  Post a comment.  Then I will help you with Question 2.  There he sample sizes are SMALL, so a "t-table" has to be used in Step 4.

 

Dr. G.

 

 

Answer 2

Paige,

 

Well, maybe there's a hiccup in how you're interpreting the problem that is getting in your way.  Rather than lay out the entire process (which you might already be reading in your textbook, or there are some great resources online that walk through the process), let me ask you something about what you've said, and then see if you need another nudge.

 

In your introduction to the problems, you said that you've got two populations.  In the first question, I see two samples:  31 males and 32 females. 

 

The whole idea behind using a t-test to compare means is that you've got two samples from the same population, and you're trying to explore whether a particular variable has an impact on the sample means.  In this case, your variable is "gender," so you've taken two samples, one where they're all males and another where they're all females.  But those aren't two different populations.  It's one population, two samples from that population.

 

Does that help you figure out which steps you'd need to follow in order to solve it?  Happy to go further if you're still stuck, but I wonder if that was the issue,

 

-- Michael

 

 

Answer 3

The trick to finding these answers is to compare sample to sample in both questions. Remember in typical problems with populations you use

test value - mean

-----------------

standard deviation

 

Well in samples, instead of standard deviation you use standard error where you divide the standard deviation by square root of sample size. In those cases you want to see how close the sample is to the population set.

 

When you have two means you want to know how close the *difference* between two means are to zero. The standard error is a little more complicated too:

http://mathbench.umd.edu/modules/statistical-tests_t-tests/page05.htm

 

So: here are the steps for either problem:

1) Calculate the standard error using the sample sizes and standard deviations for each sample.

2) calculate the difference between the means.

3) Divide 2) by 1) to find the t test value.

4) Find the degrees of freedom (sample size 1 + sample size 2 - 2)

5) Look up the t test value according to the degrees of freedom to find the critical t value.

If the test t value is greater than the critical t value there is a significant difference.