The general population (Population 2) has a mean of 30 and a standard deviation of 5, and the cutoff Z score for significance in a study involving one participant is 1.96. If the raw score obtained by the participant is 45, what decisions should be made about the null and research hypotheses?
Z = (score-mean)/SD
Calculate the Z score from your data above.
Do you think a null hypothesis can be accepted or rejected with n = 1?
Rejected
To make a decision about the null and research hypotheses, we need to compare the participant's raw score to the population mean and standard deviation.
First, let's calculate the Z-score of the participant's raw score using the formula:
Z = (X - μ) / σ
Where:
X = participant's raw score
μ = population mean
σ = population standard deviation
Given:
X = 45
μ = 30
σ = 5
Calculate the Z-score:
Z = (45 - 30) / 5
Z = 15 / 5
Z = 3
The Z-score obtained by the participant is 3.
Next, let's compare the Z-score to the cutoff Z-score for significance. The cutoff Z-score for significance in this case is 1.96.
Since the participant's Z-score (3) is greater than the cutoff Z-score for significance (1.96), it falls in the critical region and is considered statistically significant.
Based on these results, we can make the following decisions about the null and research hypotheses:
1. Null Hypothesis (H0): The null hypothesis states that there is no significant difference between the participant's score and the population mean.
Decision: Reject the null hypothesis.
2. Research Hypothesis (H1): The research hypothesis states that there is a significant difference between the participant's score and the population mean.
Decision: Accept the research hypothesis.
In summary, based on the participant's raw score and the given information, we reject the null hypothesis and accept the research hypothesis.
To make decisions about the null and research hypotheses, we need to determine whether the raw score obtained by the participant is significantly different from the general population mean.
First, let's calculate the Z-score for the participant's raw score using the formula:
Z = (X - μ) / σ
where:
Z = Z-score
X = raw score (45)
μ = population mean (30)
σ = population standard deviation (5)
Plugging in the values, we get:
Z = (45 - 30) / 5
Z = 15 / 5
Z = 3
The Z-score for the participant's raw score is 3.
Next, we compare this Z-score to the cutoff Z-score for significance, which is 1.96. If the calculated Z-score is greater than or equal to the cutoff Z-score, we can reject the null hypothesis in favor of the research hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated Z-score of 3 is indeed greater than the cutoff Z-score of 1.96. Therefore, we can reject the null hypothesis and conclude that the participant's raw score is significantly different from the general population mean.
To summarize:
- The null hypothesis is rejected.
- The research hypothesis is supported.
These decisions are based on the calculated Z-score and the cutoff Z-score for significance.