91Ó°ÊÓ

To study whether higher predictions are rated as more accurate than lower predictions, 161 college students were presented a prediction by a basketball expert about the winner of an upcoming basketball game between teams A and B. All participants learned that the expert had carefully examined the two teams' history, players, and other information. Eighty students were randomly assigned to a group in which the expert claimed the chance that A would win was \(70 \%\). The other 81 students were assigned to a group in which the expert claimed that the chance B would win was \(30 \%\). All students rated the expert on his accuracy using a scale ranging from 0 to 20 , with higher scores indicating greater accuracy. Are the conditions for two-sample \(t\) inference satisfied? (a) Maybe: the SRS condition is OK but we need to look at the data to check Normality. (b) No: scores in a range between 0 and 20 can't be Normal. (c) Yes: the SRS condition is OK and large sample sizes make the Normality condition unnecessary.

Step by step solution

01

Understand the Problem

We are presented with a problem that involves checking the conditions for two-sample t inference based on a survey given to two groups of students who rated a basketball expert's prediction accuracy. The focus is on understanding whether conditions for this statistical test are met.

02

Check SRS Condition

The problem statement indicates that each group of students was randomly assigned, implying a simple random sample (SRS) condition is satisfied for both groups. This is one of the requirements for two-sample t inference.

03

Consider Normality and Sample Size

While checking Normality usually involves looking at data distributions, in practice, the use of larger sample sizes (e.g., total sample of 161 with 80 in one group and 81 in another) allows the Central Limit Theorem to apply. This theorem states that for large sample sizes, the sampling distribution of the sample mean is approximately normal regardless of the data's distribution.

04

Evaluate the Options

Given that the SRS condition is satisfied, and the sample sizes are sufficiently large, the most appropriate choice is that the large sample sizes make the Normality condition unnecessary. Option (c) directly addresses this by stating: 'Yes: the SRS condition is OK and large sample sizes make the Normality condition unnecessary.'

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Random Sample (SRS)

A Simple Random Sample (SRS) is a foundational concept in statistics that ensures each member of a population has an equal chance of being selected. This is crucial because it reduces bias, allowing conclusions drawn from the sample to be more representative of the entire population. In the given exercise, students were randomly assigned to two different groups. This random assignment is indicative of an SRS, thereby fulfilling one of the key requirements for performing two-sample t inference.
The importance of SRS lies in its ability to provide:

• Unbiased results that are generalizable to the broader population.
• A sound basis for statistical inference because every member has an equal chance of selection.
• The ability to control for confounding variables through random assignment.

Without meeting the SRS condition, any further statistical analysis might be questionable, as bias could influence the results. Therefore, having SRS ensures that the analysis proceeds credibly for both groups.

Normality condition

The Normality condition is another important aspect of statistical inference. This condition requires that the data should, ideally, follow a normal distribution in order for certain statistical tests, like the t-test, to be valid. Practically, it is often evaluated by checking the shape of the data's distribution or by considering the sample size.
In the context of the exercise, while individual scores range between 0 and 20, the larger sample sizes significantly ease concerns about this condition. Here, instead of visually checking distribution for normality, statistical principles provide reassurance.

• If the sample size is large enough—typically over 30—deviations from normality are less alarming.
• The distribution of the sample mean will tend towards a normal distribution as sample size increases, as per Central Limit Theorem.

Thus, in large samples like the one with 161 students, the requirement for strictly normal data is relaxed, making the study's conclusions more robust and reliable.

Central Limit Theorem

The Central Limit Theorem (CLT) is a powerful tool in statistics providing insight into how distributions behave when sampling from a population. It states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the initial distribution of the population. This theorem is particularly useful in situations where normality is an initial concern.
In our scenario, even though the individual accuracy ratings might not normally distribute, the CLT suggests that with 80 students in one group and 81 in the other, both are sufficiently large to assume a normal distribution of the sample mean.

• The CLT helps justify using t-tests without strict normality by focusing on the sample mean.
• It provides confidence, through large samples, that normal approximation of the sampling distribution is valid.

By applying the CLT, researchers can confidently perform t-tests and make accurate inferences about population parameters based on sample data, even when the original data does not follow a normal distribution.