Statistics – T Tests for Means

Part 1

1. Explain how the t distribution is similar to a normal distribution, and how it difference from a normal distribution.
2. Explain the differences in terms of the null hypothesis between a one-sample t test, a two sample t test, and a matched pairs t test.
3. Suppose that you are testing the null hypothesis H₀: µ = 100 against H_a: µ < 100 based on a random sample of nine observations from a normal population. The data from the sample give a mean of 98 and a standard deviation of 3. What is the value of the t statistic?
4. In Problem 3, either find the exact p-value or determine a lower and upper bound of the p-value for the test statistic.
5. Identify the correct type of t test to use in the following situations. Choose from a 1-sample, 2-sample, or matched pairs t test.
   a) You interview 500 female students and ask each about the average number of minutes they spend using the internet each day.
   b) You interview a sample of 250 unmarried male students and 250 unmarried female students and ask each about the average number of minutes they spend using the internet each day.
   c) You interview 200 female students in their freshman year and again in their senior year and ask each about the average number of minutes they spend using the Internet each day.
6. Explain how the degrees of freedom are determined using:
   a) a one sample t test
   b) a two sample t test
7. A marketing company believes that younger adults use social media more than adults aged 35 or over. To investigate this, the company is planning to take a random sample of each group and determine the sample means.
   a) Determine which type of statistical test is appropriate.
   b) Create the appropriate null and alternate hypothesis for this test.

Part 2

1. A random sample of 25 U.S. adults were asked to estimate the average income of all U.S. households. The estimate mean was $47,000 with a standard deviation of $15,000. Construct a 95% confidence interval for this data.
2. People claim that women say more words per day than men. Estimates claim that a woman uses roughly 20,000 words per day, while a man uses approximately 7,000. To investigate this, a researcher recorded conversations of male college students over a 5-day period. The results are as follows:

 

7220                       13932                    4727                       10419

9258                        9717                    10728                       5265

12215                       9944                     7979                       12252

9307                        9086                    10780                       3357

The researcher believes that many use more than 7,000 words per day.
a) State the problem in your own words.
b) Create a plan for testing the researcher’s claim. Be sure to state the null and alternate hypotheses.
c) Carry out the appropriate hypothesis test. Begin by finding the sample mean and standard deviation. Give the value of the t statistic and give the p-value (or an estimate).
d) Formulate the statistical conclusion in terms of the null hypothesis and a practical conclusion. You may compare the p-value to 0.05 to determine if the null hypothesis should be rejected. When the p-value is less than 0.05, we reject the null hypothesis, otherwise we fail to reject the null.

3. A manufacturer states that the average lifetime of its light bulbs is 3 years. A random sample of 50 bulbs is taken and the average lifetime is found to be 34 months. The population is normal with a standard deviation of 8 months. Should the claim be rejected using the 0.05 as a benchmark for the p-value? Construct the hypothesis test for this using the four-step process. Write a detailed description of the steps: state, plan, solve, and conclude.
4. A marketing research firm wants to test whether the mean rating given by consumers to their favorite beer (denoted by X) differs from some other beer (denoted by Y). The ratings obtained from two different random groups are given below. Perform a t test to determine if there is a significant different in the mean ratings of these two beers. Use Excel to perform the calculations. See the website below for instructions.

http://www.excel-easy.com/examples/t-test.html

When explaining the test and its results, write a detailed description of the steps: state, plan, solve, and conclude. Assume unequal variances, and use 0.05 as a benchmark for the p-value.

Brand X                                   Brand Y

57                                           63

61                                           60

62                                           60

60                                           62

60                                           63

62                                           59

58                                           57

57                                           64

62                                           62

 

1. Create mock scenario and mock data for a matched pairs t test. The data set should include at least 10 pairs of data. Perform a matched pairs t test using the four-step process described above. Use Excel to perform the calculations. See the website below for instructions:

Paired Sample t Test