Question 1

A team of researchers conducted a single-sample t test and found that the 95 percent confidence interval around their sample mean includes the value of the population mean.Based on this, they should conclude that:

Accepted Answer

If the 95 percent confidence interval around the sample mean includes the value of the population mean, this means that there is not a significant difference between the sample and population means. Therefore, the null hypothesis cannot be rejected as there is not enough evidence to support the research hypothesis. The data are sufficient to assess the validity of the null hypothesis, and the conclusion is that they failed to find a significant difference between the sample and population means.

Question 2

Imagine the average time to complete a 4-year bachelor degree is actually 4.7 years based on national data.You collect data on the 68 psychology students who started school during the same semester as you, finding an average time to complete at 4.3 years with a standard deviation of 0.6 years.What would be the effect size associated with a single-sample t test?

Accepted Answer

The formula for effect size for a single-sample t test is:

d = (M - μ) / s

where d is the effect size, M is the sample mean, μ is the population mean, and s is the standard deviation of the sample.

Plugging in the values from the question, we get:

d = (4.3 - 4.7) / 0.6 = -0.67

Therefore, the effect size associated with a single-sample t test is -0.67.

Question 3

Imagine the average time to complete a 4-year bachelor degree is actually 4.7 years based on national data.You collect data on the 68 psychology students who started school during the same semester as you, finding an average time to complete at 4.3 years with a standard deviation of 0.6 years.What is your 95 percent confidence interval?

Accepted Answer

The 95% confidence interval for the mean can be calculated using the formula: mean ± (Z * (standard deviation / sqrt(n))), where Z is the Z-score corresponding to the confidence level (for 95%, Z ≈ 1.96), standard deviation is 0.6, and n is the sample size (68). Plugging in the values: 4.3 ± (1.96 * (0.6 / sqrt(68))) gives an interval of approximately [4.15, 4.45].

Question 4

Use the following to answer questions 
This figure represents the distribution of the percent of state residents living in a metropolitan area for all 50 U.S.states and Washington D.C. 
-(Figure: Percent Metropolitan) The figure indicates that:

Accepted Answer

The figure shows that the majority of states have more than 60 percent of their population living in metropolitan areas, with only a few states having less than 40 percent. Therefore, option A is the correct answer. Option B is incorrect because it assumes that each state has an equal population, whereas the figure shows the distribution of metropolitan populations among states. Option C is incorrect because the figure shows that the majority of people live in metropolitan areas, not rural areas. Option D is incorrect because the figure does not provide information on the size of urban centers within states.

Question 5

For the single-sample t test, the confidence interval is centered around the:

Accepted Answer

The confidence interval for the single-sample t test is centered around the sample mean. This means that we are estimating the population mean using the sample mean and constructing a range of values that is likely to contain the true population mean.

Question 6

When performing a single-sample t test, an effect size of 0.80 would be interpreted as a _____ effect.

Accepted Answer

An effect size of 0.80 is considered a large effect, as it indicates a substantial difference between the sample mean and the population mean. This would be significant in most contexts and would have practical implications.

Question 7

The t test is a parametric statistical test that allows one to generalize what was learned about one sample to a larger population.

Accepted Answer

The t test is a parametric statistical test that assumes the data follows a normal distribution and allows researchers to infer about the larger population based on the findings from a sample.

Question 8

The final step in conducting the single-sample t test is:

Accepted Answer

The final step in conducting the single-sample t test is to make a decision based on the calculated test statistic and the determined level of significance. This decision involves either rejecting or failing to reject the null hypothesis. Therefore, choice A is the best answer as it accurately reflects the final step in the process.

Question 9

The correct formula for the upper bound of a confidence interval for a single-sample t test is:

Accepted Answer

The correct formula for the upper bound of a confidence interval involves adding the product of the t-value and the standard error of the mean to the sample mean. This is represented in option A, where "t" stands for the t-value from the t-distribution corresponding to the desired confidence level, and "M_sample" represents the sample mean.

Question 10

A researcher collects 25 data points that yield a mean of 8.164 and a standard deviation (based on N - 1) of 2.237.If the researcher is comparing the sample to a population mean of 9.2, what is the 95 percent confidence interval?

Accepted Answer

To calculate the confidence interval, we use the formula:
CI = X̄ ± t(α/2, n-1) * (s/√n)
where X̄ is the sample mean, t(α/2, n-1) is the t-score for a given level of confidence (in this case 95% and degrees of freedom = n-1 = 24), s is the sample standard deviation, and n is the sample size.

Plugging in the given values:
X̄ = 8.164
s = 2.237
n = 25
t(α/2, n-1) = 2.064 (from t-table for degrees of freedom = 24 and 95% confidence level)

CI = 8.164 ± 2.064 * (2.237/√25) = [7.24, 9.09]

Therefore, the correct choice is B.

Question 11

When performing a single-sample t test, an effect size of 0.20 would be interpreted as a _____ effect.

Accepted Answer

An effect size of 0.20 would be interpreted as a small effect according to conventional benchmarks for effect size interpretation. Cohen's d is widely used as a measure of effect size for t-tests, and values of 0.2, 0.5, and 0.8 are often used as cutoffs for small, medium, and large effects, respectively. Therefore, a value of 0.2 falls into the small effect range.

Question 12

Two types of graphs that make it easy to see the distribution of two variables side-by-side are the:

Accepted Answer

The answer of Two types of graphs that make it...

Question 13

The correct formula for the lower bound of a confidence interval for a single-sample t test is:

Accepted Answer

The lower bound of a confidence interval is calculated by subtracting the margin of error from the sample mean, which is represented by -t(s_M) + M_sample.

Question 14

Based on the statistical results provided, which would lead us to fail to reject the null hypothesis?

Accepted Answer

In hypothesis testing, we set a significance level (α) usually at 0.05. If the p-value is less than or equal to α, we reject the null hypothesis. On the other hand, if the p-value is greater than α, we fail to reject the null hypothesis. 
In this case, option D indicates that the p-value is greater than 0.05 (p > 0.05), which implies that there is insufficient evidence to reject the null hypothesis.

Question 15

When performing a single-sample t test, an effect size of 0.50 would be interpreted as a _____ effect.

Accepted Answer

According to Cohen's d, an effect size of 0.50 is considered a medium effect.

Question 16

A researcher conducts a single-sample t test and finds statistical significance at the 0.01 level.The effect size is then calculated and found to be 0.04.What might be concluded about the findings?

Accepted Answer

The findings are statistically significant but the effect size is small, indicating it may not be practically significant. This could be due to a very large sample size.

Question 17

Which report of statistical results is in appropriate APA format?

Accepted Answer

Option C correctly follows APA format by including the t statistic value, degrees of freedom in parentheses, and the p-value, without stating a decision about the null hypothesis.

Question 18

Use the following to answer questions 
This figure represents the distribution of the percent of state residents living in a metropolitan area for all 50 U.S.states and Washington D.C. 
-(Figure: Percent Metropolitan) The plot in the figure is a:

Accepted Answer

A dot plot displays individual data points aligned along an axis, which is what the description "Percent Metropolitan" suggests, as it likely shows the distribution of some percentage data across different metropolitan areas or similar categories.

Question 19

A researcher collects 25 data points that yield a mean of 8.164 and a standard deviation (based on N - 1) of 2.237.If the researcher is comparing the sample to a population mean of 9.2 using a single-sample t test, what would be the effect size?

Accepted Answer

To calculate the effect size, we use the formula Cohen's d = (sample mean - population mean) / sample standard deviation. Plugging in the values, we get Cohen's d = (8.164 - 9.2) / 2.237 = -0.46, so the correct answer is B.

Question 20

The correct formula for effect size using Cohen's d for a single-sample t test is:

Accepted Answer

The correct formula for effect size using Cohen's d for a single-sample t test is (M - μ) / s, where M is the sample mean, μ is the population mean, and s is the sample standard deviation. The answer choice B correctly represents this formula. The other answer choices have errors in the formula or include unnecessary variables.

Quiz 9: The Single-Sample T Test