Problem 8 - An Introduction to Psychological Statistics

Question

Determine whether you would reject or fail to reject the null hypothesis in the following situations: a. \(t=2.58, N=21\), two-tailed test at \(\alpha=0.05\) b. \(t=1.99, N=49\), one-tailed test at \(\alpha=0.01\) c. \(\mu=47.82,99 \% \mathrm{CI}=(48.71,49.28)\) d. \(\mu=0,95 \% \mathrm{CI}=(-0.15,0.20)\)

Step-by-Step Solution

Answer

a. Reject, b. Fail to reject, c. Reject, d. Fail to reject.

Step 1: Determine the Critical Value for (a)

For a two-tailed test with \(\alpha = 0.05\) and \(N = 21\), we have \(df = 20\). Look up the critical \(t\)-value from the \(t\)-distribution table for \(df = 20\) and \(\alpha/2 = 0.025\). The critical value is approximately \(\pm 2.086\).

Step 2: Decision for (a)

Compare the calculated \(t\) value (2.58) with critical values \(\pm 2.086\). Since 2.58 is greater than 2.086, we reject the null hypothesis.

Step 3: Determine the Critical Value for (b)

For a one-tailed test with \(\alpha = 0.01\) and \(N = 49\), the \(df = 48\). Using the \(t\)-distribution table for \(df = 48\) at \(\alpha = 0.01\), the critical value is approximately 2.41.

Step 4: Decision for (b)

Compare the calculated \(t\) value (1.99) with the critical value 2.41. Since 1.99 is less than 2.41, we fail to reject the null hypothesis.

Step 5: Evaluate the Confidence Interval for (c)

Observe if the population mean \(\mu = 47.82\) falls within the \(99\%\) CI \((48.71, 49.28)\). Since 47.82 is outside this interval, we reject the null hypothesis.

Step 6: Evaluate the Confidence Interval for (d)

Check if the population mean \(\mu = 0\) falls within the \(95\%\) CI \((-0.15, 0.20)\). Since 0 is within this interval, we fail to reject the null hypothesis.

Key Concepts

Null Hypothesis

The null hypothesis is a foundational concept in hypothesis testing. It suggests that there is no significant effect or difference, and it serves as the default or starting assumption for testing. For instance, if you were examining whether a new drug has a different effect than an existing one, the null hypothesis would state that there is no difference in effect between the two drugs.

• In hypothesis testing, we either reject or fail to reject the null hypothesis based on statistical evidence.
• Rejecting the null hypothesis implies that we found enough statistical evidence to support the alternative hypothesis.
• Failing to reject means the evidence is insufficient to support the alternative hypothesis, so we stick with the null.

Understanding the null hypothesis is crucial because it helps frame your research questions and interpretation of results in a statistical context.

Critical Value

The critical value acts as a threshold in hypothesis testing to decide when to reject the null hypothesis. It is determined by the significance level \( \alpha \), which represents the probability of rejecting the null hypothesis when it is actually true.

• A common significance level is 0.05, but others like 0.01 or 0.10 can be used depending on the confidence required.
• The critical value is found using the t-distribution table, varying based on the degrees of freedom, which are calculated from sample size minus one.
• Compare the calculated statistic (such as a t-value) with the critical value to make your decision: if the statistic exceeds the critical value, reject the null hypothesis.

Grasping the concept of the critical value will enhance your ability to properly interpret the results of statistical tests.

Confidence Interval

A confidence interval provides a range of values, derived from the sample, that is believed to contain the population parameter with a certain level of confidence. It is an important concept that complements hypothesis testing. Here's why it matters:

• Confidence intervals quantify the uncertainty surrounding your sample estimate, providing more information than a single point estimate.
• A 95% confidence interval suggests that if the same population is sampled 100 times, approximately 95 of those confidence intervals will contain the true population parameter.
• If a hypothesized value (like a population mean under the null hypothesis) falls outside the confidence interval, you can consider rejecting the null hypothesis.

Therefore, confidence intervals not only help in decision-making but also offer insights about the precision and reliability of your estimates.

T-Distribution

The t-distribution is a key concept in statistical inference, especially when dealing with small sample sizes. It becomes essential when the population standard deviation is unknown. Here is a simplified understanding of its use:

• The t-distribution is similar to the normal distribution but has heavier tails, meaning it is more prone to producing values far away from the mean. This accounts for the uncertainty inherent in smaller samples.
• It is used to find critical values and for calculating test statistics when assessing the null hypothesis.
• As sample size increases, the t-distribution approaches the normal distribution, making it versatile for different sample sizes.

Comprehending the t-distribution assists in understanding the behavior of test statistics and the variability expected with small samples which is crucial for effective hypothesis testing.