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Need work revised and have some of the answers attached from tutor to assist 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 (highlight or use a different color).

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:
  • Before you record or analyze the data, can you assume that the sampling distribution of the mean for this test will be normally distributed?
  • 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. The height of 25 women you know has a mean of 66.84 inches. Portion of the Normal Curve Table z Area z Area z Area z Area 1.92 ..27 ..62 ..97 ..93 ..28 ..63 ..98 ..94 ..29 ..64 ..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?
  • Given your sample mean, should you consider this height unusually tall for adult women?

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

• Criterion: Differentiate between independent and dependent variables.
• Data: A researcher randomly assigns adults to two diet plans and measures weight loss over two weeks.
• Instruction:
  • What is the IV in this study?
  • 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: Studying whether weight loss differs between Diet Plan A and Diet Plan B.
• Instruction:
  • a. Write a directional alternative hypothesis.
  • b. Write a nondirectional alternative hypothesis.
  • c. Write 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: Men weigh more on average than women; a study shows no significant difference in weights between 100 men and 100 women.
• Instruction: Answer the following:
  • Given that a difference really does exist, what type of error is this? Explain.

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

• Criterion: Differentiate between Type 1 and Type 2 error.
• Data: A study finds that women score significantly higher on IQ than men, despite no true difference in the population.
• Instruction: Answer the following:
  • Identify the error (Type 1 or Type 2) and explain why.

Problem Set 3.8 : Hypothesis Testing and the z Score

• Criterion: Evaluate a null hypothesis based on data analysis.
• Data: Joan is 72 inches tall; the population mean height for women is 65 inches with SD 3.5 inches.
• Instruction:
  • a. State the null hypothesis.
  • b. State the alternative hypothesis.
  • c. What percentage of women are shorter than Joan? (Use z score and area under the curve)
  • d. Based on Joan's height, do you expect to reject the null hypothesis? Explain.

Problem Set 3.9 : One-Sample t Test

• Criterion: Hand calculate a one-sample t test.
• Data: Rose vase life days: 8, 6, 12, 11, 8, 9, 14, 15, 10; population mean is 8 days.
• Instruction: Complete:
  • a. Write the nondirectional hypothesis.
  • b. Find the critical t at α=0.05 (two-tailed).
  • c. Calculate t, showing work.
  • d. Is the vase life significantly different? Explain.

Problem Set 3.10 : One-Sample t Test in SPSS

• Criterion: Calculate and interpret a one-sample t test in SPSS.
• Data: Use the roses data from 3.9.
• Instruction: Steps:
  • a. Enter data into SPSS and name variable "Roses".
  • b. Use Analyze → Compare Means → One Sample t Test.
  • c. Send "Roses" to test variable, enter test value 8.
  • d. Run and copy output; compare to hand calculations.

Problem Set 3.11 : Confidence Intervals

• Criterion: Calculate confidence intervals from SPSS output.
• Data: Use SPSS output from 3.10 with test value of 8.
• Instruction: Calculate the 95% confidence interval based on SPSS output.

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

• Criterion: Analyze relationship between standard error and differences between means.
• Data: Two studies show same mean difference (2), but different significance due to differing standard errors.
• Instruction: Explain why the standard error differs across studies.

Problem Set 3.13 : Independent Samples t Test in SPSS

• Criterion: Conduct an independent samples t test in SPSS.
• Data: Group 1 practice with cream; Group 2 practice without. Practice minutes:
  • Group 1: 55, 44, 62, 30, 78, 50, 52
  • Group 2: 31, 40, 53, 22, 41, 16, 33
• Instruction:
  • a. Set up and enter data in SPSS with group labels.
  • b. Use Analyze → Compare Means → Independent-Samples t Test.
  • c. Send 'Minutes' to test variable; send 'Groups' as grouping variable; define groups 1 and 2.
  • d. Run and copy SPSS output.

Problem Set 3.14 : Independent Samples t Test

• Criterion: Interpret the IV, DV, hypotheses, and null hypothesis.
• Data: From 3.13; assess if practice with cream improves practice time.
• Instruction:
  • a. Identify IV and DV.
  • b. State null and alternative hypotheses (one-tailed).
  • c. Determine if you can reject null at α=0.05 and explain why.

This completes the set of problems for review and calculation. Focus on clear presentation and showing all necessary work, especially calculations.

Sample Paper For Above instruction

In this paper, I will address the essential concepts surrounding sampling, hypotheses, errors, significance testing, and confidence intervals, as outlined in the problem sets provided. I will demonstrate understanding through calculations, distinction between variables, and interpretation of statistical results, supported by scholarly references.

Introduction

Statistical analysis forms the backbone of scientific research, enabling researchers to make informed decisions about data collected from samples. Understanding the properties of sampling distributions, calculating standard errors, conducting hypothesis tests such as z and t tests, and interpreting errors are vital skills in research methodology. This paper explores these concepts through detailed responses to assigned problem sets, illustrating both theoretical and practical applications.

Central Limit Theorem and Distribution Normalcy

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, typically n>30. In the case of administering a test to 20,000 adults, the sample size far exceeds this threshold. Therefore, the sampling distribution of the mean can be assumed normal because the CLT assures normality at large sample sizes regardless of the population distribution (Shalizi, 2013). This assumption simplifies subsequent analyses, such as confidence intervals and significance testing.

Calculating the Standard Error of the Mean

The standard error of the mean (σ_M) is calculated using the formula σ_M = σ / √n, where σ is the population standard deviation and n is the sample size. Given σ=5, we find:

• a. For n=25: σ_M = 5 / √25 = 5 / 5 = 1
• b. For n=16: σ_M = 5 / √16 = 5 / 4 = 1.25
• c. For n=20: σ_M = 5 / √20 ≈ 5 / 4.4721 ≈ 1.118

These calculations demonstrate how larger samples yield smaller standard errors, increasing the precision of the sample mean estimate (Cohen, 1990).

Z Test and Probability of Tall Height

Using the data for women's heights, the z score for your friends' mean height (66.84 inches) is calculated as:

z = (X̄ - μ) / (σ/√n) = (66.84 - 65)) / (3.5 / √25) = 1.84 / (3.5 / 5) = 1.84 / 0.7 ≈ 2.63

Referring to the z table, an area to the left of z=2.63 is approximately 0.9957, meaning there is a 99.57% probability that the height of your friends falls below this value. Therefore, it is statistically uncommon but not impossible; this height is slightly above average, but not extraordinarily tall.

Independent and Dependent Variables

The independent variable (IV) in the diet study is the type of diet plan assigned (Diet Plan A or B). The dependent variable (DV) is the amount of weight loss experienced during the two weeks (Cook & Campbell, 1979). Recognizing the IV and DV clarifies the design of experimental research and helps interpret causal relationships.

Hypotheses Statements

The research hypotheses are as follows:

• a. Directional alternative hypothesis: Participants on Diet Plan A will lose more weight than those on Diet Plan B.
• b. Nondirectional alternative hypothesis: There is a difference in weight loss between Diet Plan A and B.
• c. Null hypothesis: There is no difference in weight loss between the two diet plans.

Errors in Hypothesis Testing

In the first case, failing to find a significant difference despite a true difference in the population exemplifies a Type 2 error, which occurs when the null hypothesis is incorrectly not rejected (Hancock & Algozzine, 2011). Similarly, in the second scenario, finding a significant difference where none exists is indicative of a Type 1 error, representing a false positive (Fisher, 1935).

Evaluating Joan's Height Using z Scores

Joan's z score is calculated as:

z = (72 - 65) / 3.5 ≈ 2.0

From the z table, an area to the left of z=2.0 is approximately 0.9772, meaning Joan is taller than about 97.72% of women. Since her height is significantly above average, we would reject the null hypothesis that there is no difference in heights.

One-Sample t Test for Roses' Vase Life

The sample mean is:

(8 + 6 + 12 + 11 + 8 + 9 + 14 + 15 + 10) / 9 ≈ 10.22

The sample standard deviation (s) is calculated with the formula for sample SD, resulting in approximately 3.26. The t statistic is computed as:

t = (X̄ - μ) / (s / √n) = (10.22 - 8) ) / (3.26 / √9) ≈ 2.22 / 1.088 ≈ 2.04

Critical t value at α=0.05 (two-tailed) with 8 degrees of freedom is approximately 2.306. Since 2.04

SPSS Analysis and Confidence Interval

Using SPSS with the roses data, the output should match hand calculations, providing the t statistic, degrees of freedom, and confidence intervals. The 95% confidence interval can be derived from SPSS output, typically including the sample mean ± margin of error, reflecting the range where the true population mean likely lies.

Relationship between Standard Error and Significance

In the two studies, although the mean difference is identical (2), the significance varies because the standard error differs. A smaller standard error indicates less variability in the estimate, increasing the likelihood of statistical significance, as seen in the second study. Larger standard errors imply greater variability, reducing power (Cohen, 1990).

Independent Samples t Test Interpretation

Analyzing the practice minutes in SPSS involves examining group means and t-test results, including p-values. If p

Conclusion

Understanding and correctly applying statistical concepts such as hypotheses, errors, and significance tests are essential in research. Proper calculation and interpretation ensure valid conclusions regarding data, variable relationships, and population characteristics. This analysis demonstrates practical applications of these methods, guiding future research.

References

• Cohen, J. (1990). Things I have learned (so far). American Psychologist, 45(12), 1304-1312.
• Cook, T. D., & Campbell, D. T. (1979). Quasi-experimentation: Design & analysis issues for field settings. Houghton Mifflin.
• Fisher, R. A. (1935). The design of experiments. Oliver & Boyd.
• Hancock, D., & Algozzine, B. (2011). Doing case study research: A practical guide for beginning researchers. Teachers College Press.
• Shalizi, C. R. (2013). Advanced data analysis from an elementary point of view.