PSYC2001

Question 1

What is the difference between a population and a sample?

Answer 1

A population is the entire group of interest in a study (e.g., all adults in Australia).
A sample is a smaller subset of the population that researchers study to make inferences about the whole.

Question 2

Why do researchers use samples instead of studying whole populations?

Answer 2

Populations are often too large or impractical to measure in full.
Sampling is efficient and cost-effective while still allowing for accurate conclusions if done correctly.

Question 3

How does random sampling help achieve a representative sample?

Answer 3

In random sampling, every member of the population has an equal chance of being selected.
This reduces bias and ensures the sample reflects the characteristics of the population.

Question 4

Why is convenience sampling considered biased?

Answer 4

Convenience sampling selects participants based on ease of access rather than randomness.
This can lead to a non-representative sample (e.g., only students in a psychology class).

Question 5

If a study only surveys university students, what limitations might arise?

Answer 5

University students are typically younger, more educated, and may have different socioeconomic backgrounds than the general population.
Results may not generalize to older adults, people without higher education, or different cultural groups.

Question 6

Why do researchers report sampling methods in publications?

Answer 6

Transparency about sampling methods allows readers to evaluate potential biases in the study.
It helps other researchers determine whether the findings are generalizable.

Question 7

What is the difference between descriptive and inferential statistics?

Answer 7

Descriptive statistics summarize sample data (e.g., mean, standard deviation).
Inferential statistics use sample data to make conclusions about a population (e.g., hypothesis testing).

Question 8

How does a confidence interval help in estimation?

Answer 8

A confidence interval (CI) provides a range within which the true population parameter is likely to fall.
It accounts for sampling error and indicates how precise an estimate is.

Question 9

What is the role of hypothesis testing in inferential statistics?

Answer 9

Hypothesis testing determines whether an observed effect is statistically significant or likely due to random chance.
It helps researchers confirm or reject their assumptions about a population.

Question 10

How does standard deviation differ from variance, and why is it more useful?

Answer 10

Variance (s² or σ²) measures the spread of data but is in squared units.
Standard deviation (s or σ) is the square root of variance, so it is in the same units as the data, making it more interpretable.

Question 11

What is the main goal of inferential statistics?

Answer 11

Answer: To draw conclusions about a population based on a sample and determine if observed effects are due to chance.

Question 12

What is the difference between hypothesis testing and estimation?

Answer 12

Answer: Hypothesis testing determines if an effect is statistically significant, while estimation assesses how accurately a sample statistic represents the population.

Question 13

Why do we use random sampling in psychological experiments?

Answer 13

Answer: To reduce bias and ensure that the sample is representative of the population.

Question 14

What does it mean if a result is statistically significant?

Answer 14

Answer: It means the observed effect is unlikely to have occurred by chance, suggesting a real effect in the population.

Question 15

What is the role of the p-value in hypothesis testing?

Answer 15

Answer: The p-value measures the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true.

Question 16

Scenario: A researcher tests a new cognitive therapy for depression. The experimental group has an average depression score of 15, while the control group has an average score of 18.
What steps should the researcher take to determine if this difference is meaningful?

Answer 16

Answer: Conduct a hypothesis test (e.g., t-test) to compare group means, check the p-value, and consider confidence intervals to assess the reliability of the effect.

Question 17

Scenario: You conduct a study where the treatment group shows an improvement of 3 points on a memory test, while the control group shows no change. However, your sample size is only 8 per group.
Why might the results not be conclusive?

Answer 17

Answer: Small sample sizes lead to higher sampling variability, making it harder to distinguish real effects from random noise. A larger sample would provide more reliable results.

Question 18

Scenario: A study finds that students who sleep more score 5 points higher on exams than those who sleep less. The p-value is 0.25.
Should we conclude that sleep affects exam scores? Why or why not?

Answer 18

Answer: No, because a p-value of 0.25 suggests that the observed difference is likely due to chance (typically, p < 0.05 is considered significant).

Question 19

What is the sampling distribution?

Answer 19

Answer: A hypothetical distribution of sample statistics obtained by repeatedly sampling from a population.

Question 20

Why do sample means differ from the population mean?

Answer 20

Answer: Due to sampling variability, which happens because each random sample is slightly different from the population.

Question 21

How does sample size (N) affect the sampling distribution?

Answer 21

Answer: Larger sample sizes reduce sampling variability, making the sample mean a more accurate estimate of the population mean.

Question 22

Why don’t researchers take multiple samples in real experiments?

Answer 22

Answer: It’s too time-consuming, costly, and impractical. Instead, we use statistical theory and simulations to estimate sampling distributions.

Question 23

How does the sampling distribution help in hypothesis testing?

Answer 23

Answer: It allows us to determine whether an observed sample mean is likely due to random chance or represents a real effect.

Question 24

Scenario: A researcher wants to estimate the average IQ of university students. The population IQ is unknown. They collect a sample of 50 students and find a mean IQ of 108.
What concept explains why this sample mean might not be exactly the true population mean?

Answer 24

Answer: Sampling variability—each sample drawn from a population will have some natural fluctuation.

Question 25

Scenario: Suppose you conduct a study and obtain a sample mean of 75. A friend runs the same study with a different sample and gets a mean of 78. What does this tell you about sampling distributions?

Answer 25

Answer: This variability is expected—sample means fluctuate around the true population mean. If we repeated the study many times, we'd see a distribution of sample means.

Question 26

Scenario: A psychologist collects a sample of 100 participants and finds a mean stress score of 50. Another psychologist collects a sample of only 10 participants and finds a mean stress score of 60. Which result is likely more reliable, and why?

Answer 26

Answer: The result from the larger sample (N = 100) is more reliable because larger samples reduce sampling variability, making estimates closer to the true population mean.

Question 27

What two factors influence the variance of a sampling distribution?

Answer 27

Answer: The population variance (σ²) and the sample size (N).

Question 28

What is the standard error of the mean (SEM)?

Answer 28

Answer: The standard deviation of the sampling distribution, calculated as σN\frac{\sigma}{\sqrt{N}}N​σ​. It quantifies how much sample means vary from the true population mean.

Question 29

How does sample size affect the sampling distribution?

Answer 29

Answer: Larger sample sizes reduce the variability of sample means, making the sampling distribution narrower and more accurate in estimating the population mean.

Question 30

What does the Central Limit Theorem (CLT) state about the sampling distribution of the mean?

Answer 30

Answer: Regardless of the shape of the population distribution, the sampling distribution of the mean will tend towards a normal distribution as the sample size increases.

Question 30

Scenario: A population has a high variance (σ² = 100). You conduct two studies: one with a sample size of 25, and another with a sample size of 100. Question: Which sampling distribution will be narrower and why?

Answer 30

Answer: The sampling distribution from the sample size of 100 will be narrower because a larger sample reduces variability in sample means, making them more accurate estimates of the population mean.

Question 31

Scenario: You are working with a population that has a skewed distribution (many low values and a few very high values). You take two samples: one of size 10 and the other of size 50. Question: What will the shape of the sampling distribution of the mean look like for each sample size?

Answer 31

Answer: For the smaller sample (N = 10), the sampling distribution of the mean might not be normal and may reflect the skewed population. However, for the larger sample (N = 50), the sampling distribution of the mean will likely approximate a normal distribution due to the Central Limit Theorem.

Question 32

Scenario: A population has a normal distribution, and you collect a sample of size 25. Question: What will the sampling distribution of the sample mean look like?

Answer 32

Answer: The sampling distribution of the mean will also be normal, since the population is normal, and the sample size is sufficiently large (25 ≥ 30 is often used, but this is a close enough approximation)

Question 33

What is the standard error of the mean (SEM)?

Answer 33

Answer: It is the standard deviation of the sampling distribution of the sample mean, calculated as: σM=σN\sigma_M = \frac{\sigma}{\sqrt{N}}σM​=N​σ​

Question 34

What does the area under the normal curve represent?

Answer 34

Answer: It represents the probability of obtaining a sample mean within a given range.

Question 35

Why is the sample mean (M) the best point estimate for μ?

Answer 35

Answer: Because it is an unbiased estimator, meaning its expected value equals μ.

Question 36

What does a confidence interval tell us?

Answer 36

Answer: It gives a range of values within which the true population mean (μ) is likely to be, with a certain confidence level (e.g., 95%).

Question 37

What happens to the confidence interval if the sample size increases?

Answer 37

Answer: The confidence interval becomes narrower because the standard error decreases.

Question 38

A population has μ = 120 and σ = 20. If we take a sample of size N = 16, what is the standard error of the mean (SEM)?

Answer 38

σM=2016=204=5\sigma_M = \frac{20}{\sqrt{16}} = \frac{20}{4} = 5σM​=16​20​=420​=5

Question 39

Using the same data (μ = 120, σ = 20, N = 16), what is the 95% confidence interval if M = 125?

Answer 39

Standard error: 5 Confidence interval formula: 125±(1.96×5)125 \pm (1.96 \times 5)125±(1.96×5) Calculation: 125±9.8=(115.2,134.8)125 \pm 9.8 = (115.2, 134.8)125±9.8=(115.2,134.8) Interpretation: We are 95% confident that μ is between 115.2 and 134.8.

Question 40

Question: A researcher conducts a study on sleep duration and finds that the mean sleep time in a sample of university students is 6.5 hours, with a standard error of 0.4 hours. The 95% confidence interval is (5.7, 7.3) hours. What does this mean in the context of the study?

Answer 40

Answer: This means that if we repeatedly sampled university students and calculated confidence intervals each time, 95% of those intervals would contain the true mean sleep duration of the population. However, we do not say there is a 95% probability that the true mean is in (5.7, 7.3)—the true mean is either inside or outside, but we don’t know for sure.

Question 41

Question: A company tests a new drug that claims to reduce anxiety. The sample mean anxiety score after treatment is 30, with a 95% confidence interval of (25, 35). If the known population mean anxiety score without treatment is 37, should the company conclude that the drug is effective?

Answer 41

Answer: Yes. Since the population mean 37 is outside the confidence interval (25, 35), this suggests that the drug significantly reduces anxiety. If the true mean anxiety score was still 37, it would be expected to fall inside most confidence intervals. Because it does not, we can infer a significant effect of the drug.

Question 42

Question: Two researchers conduct the same study but with different sample sizes: Researcher A: N = 25, 95% CI = (50, 60) Researcher B: N = 100, 95% CI = (54, 56) Why is Researcher B’s confidence interval narrower?

Answer 42

Answer: A larger sample size (N = 100) reduces the standard error, making the confidence interval smaller and more precise. Researcher A has a wider confidence interval because their smaller sample leads to greater variability.

Question 43

What is the null hypothesis in hypothesis testing?

Answer 43

Answer: The null hypothesis (H₀) states that there is no effect or no difference (e.g., μ = 50).

Question 44

Why do we use a significance level (α) of 0.05?

Answer 44

Answer: It balances the risk of Type I and Type II errors, ensuring that false positives occur only 5% of the time.

Question 45

What does rejecting the null hypothesis mean?

Answer 45

Answer: It means the sample provides sufficient evidence that μ is different from μ₀, but it does not prove μ = M exactly.

Question 46

Why do we report the direction of results?

Answer 46

Answer: To avoid misinterpretation (e.g., stating a drug is effective without specifying it worsens symptoms).

Question 46

What is the difference between a Type I and Type II error?

Answer 46

Type I Error: Rejecting a true null hypothesis (false positive). Type II Error: Failing to reject a false null hypothesis (false negative).

Question 47

A pharmaceutical company tests a new drug. The null hypothesis is that the drug has no effect on blood pressure (H₀: μ = 120). The alternative hypothesis is that the drug changes blood pressure (H₁: μ ≠ 120). A sample of 50 patients shows an average BP of 115 with Z = -2.3 (critical value = ±1.96). Q1: Should the null hypothesis be rejected? Q2: What type of error could occur if the null is rejected incorrectly?

Answer 47

Q1: Should the null hypothesis be rejected? A1: Yes, because |Z| = 2.3 > 1.96, meaning there is significant evidence that the drug affects BP. Q2: What type of error could occur if the null is rejected incorrectly? A2: A Type I error (false positive)—the drug might not actually affect BP, but we wrongly conclude that it does.

Question 48

A researcher compares literacy scores of 5th graders in Australia and the U.S. The null hypothesis states no difference (H₀: μ_AUS = μ_US = 50). The researcher finds M_AUS = 53, Z = 1.5. Q3: Should the null hypothesis be rejected? Q4: What does this conclusion mean? Q5: What type of error is possible here?

Answer 48

Q3: Should the null hypothesis be rejected? A3: No, because |Z| = 1.5 < 1.96, meaning the result is not statistically significant at α = 0.05. Q4: What does this conclusion mean? A4: It means there isn’t enough evidence to say Australian students have different literacy levels than U.S. students. Q5: What type of error is possible here? A5: A Type II error (false negative)—there could be a real difference, but the test failed to detect it.

Question 49

A study tests a weight-loss program. The null hypothesis states it has no effect (H₀: μ = 0 kg lost). After 100 participants, the study finds an average weight loss of 2.1 kg, Z = 2.8. Q6: What is the conclusion? Q7: If α had been set at 0.01 instead of 0.05, what would happen?

Answer 49

Q6: What is the conclusion? A6: Reject H₀ because Z = 2.8 > 1.96. There is significant evidence that the program leads to weight loss. Q7: If α had been set at 0.01 instead of 0.05, what would happen? A7: The new critical value would be ±2.576. Since 2.8 > 2.576, the result would still be significant.

Question 50

What is the difference between estimation and hypothesis testing?

Answer 50

Estimation provides a range (confidence interval) for a population parameter. Hypothesis testing determines whether sample data provides enough evidence to reject a null hypothesis.

Question 51

Why do we start with a null hypothesis instead of assuming the alternative is true?

Answer 51

Because it's a conservative approach: we assume no effect unless there’s strong evidence against it.

Question 52

What does it mean when we set α = 0.05?

Answer 52

There is a 5% chance of incorrectly rejecting the null hypothesis (Type I error).

Question 53

If a p-value is 0.03, what should you do at a significance level of 0.05?

Answer 53

Reject the null hypothesis because 0.03 < 0.05, meaning the result is statistically significant.

Question 54

What are Type I and Type II errors? Give an example of each.

Answer 54

Type I Error: Rejecting H₀ when it is actually true (e.g., convicting an innocent person). Type II Error: Failing to reject H₀ when it is false (e.g., letting a guilty person go free).

Question 55

A new drug is tested to see if it lowers blood pressure. The null hypothesis states the drug has no effect (H₀: µ = 120 mmHg). The alternative hypothesis states the drug reduces blood pressure (H₁: µ < 120 mmHg). If the drug group’s mean blood pressure is 118 mmHg with p = 0.04, what do you conclude at α = 0.05? If the drug group’s mean blood pressure is 119 mmHg with p = 0.07, what do you conclude at α = 0.05?

Answer 55

If the drug group’s mean blood pressure is 118 mmHg with p = 0.04, what do you conclude at α = 0.05? Answer: Reject H₀, conclude the drug significantly lowers blood pressure. If the drug group’s mean blood pressure is 119 mmHg with p = 0.07, what do you conclude at α = 0.05? Answer: Retain H₀, not enough evidence to say the drug works.

Question 56

A company tests two website versions: H₀: The new version does not increase sales. H₁: The new version increases sales. If p = 0.02, what should the company do? If p = 0.10, what should the company do?

Answer 56

If p = 0.02, what should the company do? Answer: Reject H₀, conclude the new version improves sales. If p = 0.10, what should the company do? Answer: Retain H₀, insufficient evidence to conclude the new version is better.

Question 57

Why do we divide by (n-1) instead of n when estimating variance from a sample?

Answer 57

To correct for bias in underestimating the population variance.

Question 58

What is the t-distribution, and why do we use it instead of the normal distribution?

Answer 58

The t-distribution accounts for additional variability from estimating σ. It is used when σ is unknown.

Question 59

How does the t-distribution change as sample size increases?

Answer 59

It becomes more similar to the normal distribution as df increases.

Question 60

How can confidence intervals be used to interpret hypothesis tests?

Answer 60

If µ0 is inside the confidence interval, we fail to reject H0. If µ0 is outside, we reject H0.

Question 61

What does it mean if a hypothesis test is statistically significant?

Answer 61

It means that the difference between M and µ0 is unlikely to be due to chance, but it does not indicate effect size.

Question 62

A sample of n = 9 has a mean of M = 50 and s = 6. Compute the 95% confidence interval for µ.

Answer 62

Solution: df = 9 - 1 = 8 tc from t-table (df = 8, α = 0.05): 2.306 Standard error: sM=69=2s_M = \frac{6}{\sqrt{9}} = 2sM​=9​6​=2 Confidence interval: μupper=50+(2.306×2)=54.61\mu_{\text{upper}} = 50 + (2.306 \times 2) = 54.61μupper​=50+(2.306×2)=54.61 μlower=50−(2.306×2)=45.39\mu_{\text{lower}} = 50 - (2.306 \times 2) = 45.39μlower​=50−(2.306×2)=45.39 Final answer: (45.39, 54.61)

Question 63

A university claims students study an average of 15 hours per week. A sample of 25 students has M = 17 and s = 4. Test whether the university’s claim is true at α = 0.05 (two-tailed test).

Answer 63

H0: µ = 15, H1: µ ≠ 15 df = 25 - 1 = 24 tc from t-table (df = 24, α = 0.05): 2.064 Standard error: sM=425=0.8s_M = \frac{4}{\sqrt{25}} = 0.8sM​=25​4​=0.8 t-value: t=17−150.8=20.8=2.5t = \frac{17 - 15}{0.8} = \frac{2}{0.8} = 2.5t=0.817−15​=0.82​=2.5 Decision: |2.5| > 2.064, so we reject H0. Conclusion: There is significant evidence that the mean study time differs from 15 hours.

Question 64

A coffee shop surveys 16 customers and finds an average rating of 8.2 out of 10 with a sample standard deviation of 1.5. The manager claims the true satisfaction rating is 8.5. At α = 0.05, can we conclude customers are less satisfied than the manager claims?

Answer 64

H0: µ = 8.5 (satisfaction is as claimed) H1: µ < 8.5 (customers are less satisfied) df = 16 - 1 = 15 tc from t-table (df = 15, one-tailed test, α = 0.05): 1.753 sM = 1.5 / √16 = 0.375 t-value = (8.2 - 8.5) / 0.375 = -0.8 Decision: |-0.8| < 1.753, so we fail to reject H0. Conclusion: Not enough evidence to claim customers are less satisfied.

Question 65

A teacher believes students study at least 10 hours for their final exams. A random sample of 30 students reports an average study time of 9.5 hours, with a standard deviation of 2 hours. Can we conclude students study less than 10 hours?

Answer 65

Solution: H0: µ = 10, H1: µ < 10. df = 30 - 1 = 29. tc from t-table (df = 29, one-tailed test, α = 0.05): 1.699. sM = 2 / √30 = 0.365. t-value = (9.5 - 10) / 0.365 = -1.37. Decision: |-1.37| < 1.699, so we fail to reject H0. Conclusion: Not enough evidence to prove students study less than 10 hours.

Question 66

What is the difference between a paired samples and an independent samples design?

Answer 66

Paired samples involve related groups (e.g., repeated measures), while independent samples consist of separate, unrelated groups.

Question 67

What does the null hypothesis (H₀) state in a paired samples t-test?

Answer 67

H₀: The mean difference between the two conditions is zero (μD = 0).

Question 68

When do we reject H₀ in a paired samples t-test?

Answer 68

When |t| is greater than the critical t-value or when p ≤ α.

Question 69

Why do we use difference scores in a paired samples t-test?

Answer 69

To reduce two related scores into a single score, simplifying analysis.

Question 70

Scenario 1: A psychologist tests participants' anxiety levels before and after a mindfulness program. What type of test should they use?

Answer 70

A paired samples t-test because the same participants are tested before and after the program.

Question 71

Scenario 2: Researchers measure students' test performance before and after using a new study method. The mean difference in scores is 5 points, with a p-value of 0.03. Should they reject H₀ at α = 0.05?

Answer 71

Yes, because p = 0.03 is less than 0.05, indicating a significant difference.

Question 72

Scenario 3: A company tests two marketing campaigns by exposing one group of customers to Campaign A and another to Campaign B. Should they use a paired samples t-test?

Answer 72

No, because the two groups are independent. An independent samples t-test is appropriate.

Question 73

What is the difference between a paired samples and an independent samples design?

Answer 73

Paired samples involve related groups (e.g., repeated measures), while independent samples consist of separate, unrelated groups.

Question 74

What does the null hypothesis (H₀) state in a paired samples t-test?

Answer 74

H₀: The mean difference between the two conditions is zero (μD = 0).

Question 75

When do we reject H₀ in a paired samples t-test?

Answer 75

When |t| is greater than the critical t-value or when p ≤ α.

Question 76

Why do we use difference scores in a paired samples t-test?

Answer 76

To reduce two related scores into a single score, simplifying analysis.

Question 77

Scenario 1: A psychologist tests participants' anxiety levels before and after a mindfulness program. What type of test should they use?

Answer 77

A paired samples t-test because the same participants are tested before and after the program.

Question 78

Scenario 2: Researchers measure students' test performance before and after using a new study method. The mean difference in scores is 5 points, with a p-value of 0.03. Should they reject H₀ at α = 0.05?

Answer 78

Yes, because p = 0.03 is less than 0.05, indicating a significant difference.

Question 79

Scenario 3: A company tests two marketing campaigns by exposing one group of customers to Campaign A and another to Campaign B. Should they use a paired samples t-test?

Answer 79

No, because the two groups are independent. An independent samples t-test is appropriate.