**u03a1** **–** **Sampling, Hypotheses, Errors, Significance,** **One-Sample** **t****Tests, Independent Samples** **t****Tests,** **and Confidence Intervals**

Complete the following problems within the Word document (do not submit other files). Show your work for problem sets that require calculations. Ensure that your answer to each problem is clearly visible.(You may want to highlight your answer or use a different type of color to set it apart.)

**Problem** **Set 3.1**: **Central Limit** **Theorem**

• **Criterion**: Determine if a distribution will be normally distributed based on sample size.• **Data**: As part of a large research study, you administer a new test to 20,000 adults.• **Instruction**: Answer the following:o Before you record or analyze the data, can you assume that the sampling distribution of the mean for this test will be normally distributed?o Why or why not?

**Problem** **Set 3.2**: **Standard Error of the Mean**

• **Criterion**: Calculate the standard error of the mean.• **Data**: College students in a large psychology class take a final exam. The mean exam score is 85, and the standard deviation is 5.• **Instruction**: Using the formula for *σ*M, what will the standard error of the mean (*σ*M ) be when:

a. The sample size is 25.

b. The sample size is 16.

c. The sample size is 20.

**Problem** **Set 3.3**: **z****Test**

• **Criterion**: Calculate a *z* test to make a decision about a sample.• **Data**: The average (mean) height for adult women is 65 inches, and the standard deviation is 3.5 inches. Given the women you know, you think this number is low; so, you record the heights of 25 of your female friends. The average height of your 25 friends is 66.84 inches.

**Portion of the Normal Curve Table**

**z**

**Area**

**z**

**Area**

**z**

**Area**

**z**

**Area**

1.92

.9726

2.27

.9884

2.62

.9956

2.97

.9985

1.93

.9732

2.28

.9887

2.63

.9957

2.98

.9986

1.94

.9738

2.29

.9890

2.64

.9959

2.99

.9986

• **Instruction**: Answer this: If your friends are just a representative sample of adult females, what is the probability that your friends are so tall?

*(Assessment* *continues on next page.)*

**Problem** **Set 3.4**: **Independent Variables (IVs) and Dependent Variables (DVs)**

• **Criterion**: Differentiate between independent and dependent variables.• **Data**: A researcher randomly assigns a group of adults to one of two diets plans (Diet Plan A or Diet Plan B), and she then measures the amount of weight loss each experiences in a two-week period.• **Instruction**: Answer the following: o What is the IV in this study?o What is the DV in this study?

**Problem** **Set 3.5**: **Hypotheses**

• **Criterion**: Write a directional-alternative hypothesis, nondirectional-alternative hypothesis, and null hypothesis.• **Data**: A researcher is studying whether the amount of weight loss differs in participants who follow Diet Plan A versus those who follow Diet Plan B.• **Instruction**: Write the following:

a. A directional alternative hypothesis.

b. A nondirectional alternative hypothesis.

c. The null hypothesis.

**Problem** **Set 3.6**: **Errors and Significance: Type 1 and Type 2** **Error**

• **Criterion**: Differentiate between Type 1 and Type 2 error.• **Data**: It is an established fact that in the general population, men weigh more on average than women. In a study, you randomly sample 100 men and 100 women, you record each participant’s weight, and you find that there is no significant difference in weight based on gender.• **Instruction**: Answer the following**:**o Given that a difference really does exist in the population, what type of error is this (Type 1 or Type 2)? Explain your answer.

**Problem** **Set 3.7**: **Errors and Significance: Type 1 and Type 2** **Error**

• **Criterion**: Differentiate between Type 1 and Type 2 error.• **Data**: In general, men and women do not differ on IQ. However, as part of your study, you found that women scored significantly higher than men on IQ.• **Instruction**: Answer the following**:**o Given that you found a difference in your study where none exists in the general population, identify the error (Type 1 or Type 2) and explain your answer.

*(Assessment* *continues on next page.)*

**Problem** **Set 3.8**: **Hypothesis Testing and the** **z****Score.**

• **Criterion**: Evaluate a null hypothesis based on analysis of data.• **Data**: Joan is 72 inches tall. The average (mean) height for adult women is 65 inches, and the standard deviation is 3.5 inches**.**• **Instructions**:

a. State the null hypothesis.

b. State the alternative hypothesis.

c. Answer this: Joan is taller than what percentage of women in the population? (*Hint:* Think *z* score and area under the normal curve.)

d. Answer this: Given Joan’s height as compared to the population mean, do you expect to reject the null hypothesis? Explain your answer.

**Problem Set 3.9**: **One-Sample** **t****Test**

**Criterion**: Hand calculate a one-sample *t* test.

**Data**: Rex’s Flower Shop advertised fresh cut roses that last longer than other roses. The mean vase life for a rose is 8 days. The following is a sample of the number of vase life days for 9 bouquets of roses from Rex’s Flower Shop: 8, 6, 12, 11, 8, 9, 14, 15, 10.

**Instruction**: Complete the following:

a. State the nondirectional hypothesis.

b. State the critical *t* for *α* = .05 (two tails).

c. Calculate *t.* Show your work.

d. Answer: Is the vase life of Rex’s roses significantly different than the population mean? Explain.

**Remember, you must show all of your work to receive credit.**

**Problem Set 3.10**: **One-Sample** **t****Test in SPSS**

**Criterion**: Calculate a one-sample *t* test in SPSS.

**Data**: Use the Rex’s Flower Shop data from Problem Set 3.9.

**Instruction**: Complete the following:

a. Enter the data from Problem Set 3.9 into SPSS and name the variable as **Roses**.b. In the **Toolbar**, click **Analyze**, select **Compare Means**, and then select **One**–**Sample** **t****Test**.c. Select **Roses** and then click **Arrow** to send it over to the right side of the table. In the box labeled **Test Value**, enter **8**.

*(Assessment* *continues on next page.)*

d. Click **OK** and copy and paste the output into this Word document.e. Compare your SPSS output to your hand calculations from Problem Set 3.9. Are they the same?

**Problem Set 3.11**: **Confidence Intervals**

**Criterion**: Calculate confidence intervals using SPSS.

**Data**: Use the SPSS output from Problem Set 3.10.

**Instruction**: Based on the SPSS output fom 3.10 including a test value (population mean) of 8, calculate the 95 percent confidence interval.

**Problem** **Set 3.12**: **Standard Error of the Difference Between the Means**

**Criterion**: Analyze the relationship between standard error and thedifference between the means.

**Data**: A researcher examines the results of two separate studies. In the first study, the difference between Group A and Group B is two points but the standard error is large and the difference is not significant. In the second study, the difference between Group A and Group B is also two points but the standard error is small and the difference is significant.

**Instruction**: Answer this: What might be the reason for the difference in the standard error across the two studies?

**Problem Set 3.13**: **Independent Samples** **t****Test in SPSS**

• **Criterion**: Calculate an independent samples *t* test in SPSS.• **Data**: Ms Z has two groups of band students. She asks Group 1 to use her new embouchure strengthening cream before practice and has Group 2 practice as usual. The groups scored as following:o Minutes of practice:▪ Group 1: 55, 44, 62, 30, 78, 50, 52.▪ Group 2: 31, 40, 53, 22, 41, 16, 33.• **Instruction**: Complete the following steps:a. Open the SPSS and create a **New DataSet**.b. Click the **Variable View** tab and type **Groups** in the **Name**column. Click on the gray box in the **Values** column. **Value Labels** window appears. Enter **1** in the **Value** area and enter **Embouchure Cream** in the **Label** area. Click **Add**. Now enter **2** in the **Value** area and enter **No Cream** in the **Label** area. Click **Add**. Click **OK**. The **Variable View** screen appears.

*(Assessment* *continues on next page.)*

c. In row two, enter **Minutes** in the **Name** column.d. Click **Data View**.e. Enter the minutes of practice data (For example, 1 under Groups and 55 under **Minutes**; 2 under Groups and 31 under **Minutes**).f. In the **Toolbar**, click **Analyze**, select **Compare Means**, and then select **Independent-Samples** **t****Test.**g. Select **Minutes** and then click **Arrow** to send it over to the **Test Variable** box.h. Select **Groups** and then click **Arrow** to send it over to the **Grouping Variable** box.i. Click **Define Groups** and enter **1** for Group 1 and enter **2** for Group 2. Click **Continue**.j. Click **OK** and then copy and paste the output to the Word document.

**Problem Set 3.14**: **Independent Samples** **t****Test**

**Criterion**: Identify IV, DV, and hypotheses, and evaluate the null hypothesis for an independent samples *t* test.

**Data**: Use the information from Problem Set 3.13.

**Instruction**: Complete the following:

a.Identify the IV and DV in this study.

b.State the null hypothesis and the directional (one-tailed) alternative hypothesis.

c.Can you reject the null hypothesis at α = .05? Explain why or why not.

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