Suppose A Professor Splits Their Class Into Two Groups

Suppose a professor splits their class into two groups: students whose last names begin with A-K and students whose last names begin with L-Z. If p1 and p2

Suppose a professor splits their class into two groups: students whose last names begin with A-K and students whose last names begin with L-Z. If p1 and p2 represent the proportion of students who have an iPhone by last name, would you be surprised if p1 did not exactly equal p2? If we conclude that the first initial of a student's last name is NOT related to whether the person owns an iPhone, what assumption are we making about the relationship between these two variables?

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The scenario presented by the professor involves dividing a class into two groups based on the initial letters of students' last names, with the aim of analyzing whether this categorization correlates with iPhone ownership. The core statistical question is whether the proportions of students owning an iPhone, denoted by p₁ and p₂, are significantly different between these two groups. Given naturally that variation exists in any sample, it would generally not be surprising if p₁ does not exactly equal p₂, since perfect equality between sample proportions is highly unlikely in practice. Instead, the critical consideration is whether any observed difference is statistically significant or could be attributed to random variation.

The second part of the question explores the assumption regarding the relationship between the first initial of a student's last name and iPhone ownership. If we conclude that the first initial is NOT related to ownership of an iPhone, this implies an underlying assumption of independence between these two variables. In statistical terms, independence means that knowing the initial letter of a student's last name provides no information about whether they own an iPhone. This assumption allows researchers to treat the two variables as unrelated, meaning that the distribution of iPhone ownership should be similar across the two groups if the initial letter has no impact.

This assumption of independence is fundamental in many statistical analyses because it justifies using methods such as hypothesis testing for differences in proportions. If the variables are truly independent, then any observed differences are likely due to chance rather than a causal relationship. Conversely, if the variables were related, this would suggest some underlying factor that links last name initials to ownership behavior, which would violate the assumption of independence and require different analytical approaches.

In summary, expecting p₁ and p₂ to be equal is not necessary; variability is natural in sampling. The critical assumption when concluding that the first initial of the last name is not related to iPhone ownership is that the two variables are independent, meaning the initial letter has no bearing on whether a student owns an iPhone. This assumption supports inference that any differences in proportions are due to random variation rather than an actual relationship.

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