. Each of the following sets of sample statistics comes from a within- subjects design.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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#### Source: Lab Manual for Statistical Analysis — Dawn M. McBride, J. Cooper Cutting
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#### Table: Handedness Scores of Twin Pairs
| Pair | Twin A | Twin B |
|------|--------|--------|
| 1 | +10 | +11 |
| 2 | -8 | +3 |
| 3 | -11 | +11 |
| 4 | +15 | +10 |
| 5 | 0 | +8 |
| 6 | -4 | +7 |
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#### Analysis Task
4. **Question:**
Each of the following sets of sample statistics comes from a within-subjects design.
**Set 1:**
- \( n = 10 \)
- \( \overline{D} = +4.0 \)
- \( s = 10 \)
**Set 2:**
- \( n = 10 \)
- \( \overline{D} = +4.0 \)
- \( s = 2 \)
**Task:**
Find t values. Without looking up the critical t, for which set is it more likely to reject the \( H_0 \) indicating that the \( \mu_D = 0 \)? Why?
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### Section 32: Related Samples T Test in SPSS
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#### Explanation:
The table above presents handedness scores for twin pairs, indicating each twin's score for six pairs. These scores can be analyzed for statistical differences within pairs. The related samples t-test is used to determine if there is a significant difference between the means of two related groups. In this case, those groups are the scores of Twin A and Twin B across pairs.
The task given involves two sets of sample statistics to assess which is more likely to reject the null hypothesis \( H_0 \), assuming \( \mu_D = 0 \) (the population mean difference is zero). The decision is based on comparing the mean differences and standard deviations across the sample sizes provided.
Set 1 and Set 2 both have the same mean difference but different standard deviations, with Set 2 having a smaller standard deviation, which makes it more likely to achieve statistical significance given the same size \( n \).
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