91Ó°ÊÓ

At a university, \(60 \%\) of the 7,400 students are female. The student newspaper reports results of a survey of a random sample of 50 students about various topics involving alcohol abuse, such as participation in binge drinking. They report that their sample contained 26 females. a. Explain how you can set up a binary random variable \(X\) to represent gender. b. Identify the population distribution of gender at this university. Sketch a graph. c. Identify the data distribution of gender for this sample. Sketch a graph. d. Identify the sampling distribution of the sample proportion of females in the sample. State its mean and standard deviation for a random sample of size \(50 .\) Sketch a graph. e. Use the Sampling Distribution app accessible from the book's website to check your answers from parts b through d. Set the population proportion equal to \(p=0.60\) and \(n=50 .\) Compare the population, data, and sampling distribution graph from the app with your graphs from parts b through d.

Step by step solution

01

Define the Binary Random Variable

To set up a binary random variable \(X\) to represent gender, we can assign values to the two possible outcomes. Let's define \(X = 1\) if the student is female and \(X = 0\) if the student is male. This way, \(X\) represents the gender of a randomly selected student from the university.

02

Determine the Population Distribution

The population distribution of gender at the university is determined by the overall proportion of female and male students. Given that 60% of the students are female, the probability distribution is as follows:- \(P(X = 1) = 0.60\)- \(P(X = 0) = 0.40\)This can be represented graphically with a bar graph showing two bars: one for 0.60 (females) and one for 0.40 (males).

03

Determine the Data Distribution of the Sample

In the sample of 50 students, there are 26 females. Thus, the sample proportion \(\hat{p}\) of females is \(\hat{p} = \frac{26}{50} = 0.52\). The data distribution for this sample is a discrete distribution with only two outcomes:- \(P(X = 1) = 0.52\) (for females)- \(P(X = 0) = 0.48\) (for males)Graphically, represent it with two bars: one for 0.52 (females) and one for 0.48 (males).

04

Identify the Sampling Distribution

The sampling distribution of the sample proportion is based on the assumption of a binomial distribution. For a random sample of size 50 and population proportion \(p = 0.60\), the sampling distribution of the sample proportion \(\hat{p}\) is approximately normal due to the Central Limit Theorem.- Mean \( \mu_{\hat{p}} = p = 0.60 \)- Standard deviation \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.60 \times 0.40}{50}} = 0.0693 \)Sketch this distribution as a normal curve centered at 0.60 with a standard deviation of 0.0693.

05

Verify Using Sampling Distribution App

Use a Sampling Distribution app to input the population proportion \(p = 0.60\) and sample size \(n = 50\). Observe the graphs produced. Compare the population distribution graph (60% female, 40% male) with the sample's data distribution of 52% female. Also, compare the sampling distribution graph centered at 0.60 with our calculated values. Ensuring the graphs align will verify the solution steps.

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Key Concepts

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

Binary Random Variable

In statistics, a binary random variable is used to represent two possible outcomes of an event. Here, the task is to model gender as a binary random variable called \(X\). This means we set \(X = 1\) if the student is female and \(X = 0\) if the student is male.

• Binary random variables often simplify the analysis of data with two categories.
• They are like a switch that is either on (1) or off (0) and help to keep things straightforward, especially when dealing with large data sets.

The variable \(X\) encapsulates the gender of randomly chosen students, giving us a framework to explore data collection and probabilistic inference.

Population Distribution

The population distribution provides insight into the overall makeup of a group. At the university, 60% of the students are female, forming the population distribution.

• The probability that a student chosen randomly is female is \(P(X = 1) = 0.60\).
• Conversely, the probability of selecting a male student is \(P(X = 0) = 0.40\).

This information can be visualized through a bar graph with two bars: one representing females at 0.60 and the other males at 0.40. Such visualization aids in interpreting statistical data easily, highlighting the disparities or equalities within a population.

Sampling Distribution

The concept of sampling distribution is crucial in making inferences about the population from a sample. In this scenario, the sample proportion of females, \(\hat{p}\), is 0.52 because 26 out of 50 surveyed students are female.

• This distribution typically derives from a binomial distribution, but the Central Limit Theorem allows us to approximate it to a normal distribution because our sample size is sufficiently large.
• The mean of the sampling distribution, \(\mu_{\hat{p}}\), is the population proportion, which is 0.60.
• Its standard deviation, \(\sigma_{\hat{p}}\), is calculated as \( \sqrt{\frac{0.60 \times 0.40}{50}} = 0.0693 \).

Graphically, the sampling distribution can be represented as a normal curve centered around the population proportion, helping us understand variability and reliability of sample data in estimating population parameters.

Central Limit Theorem

The Central Limit Theorem (CLT) is a foundational principle in statistics that allows us to understand the behavior of sample means or proportions. According to CLT:

• Regardless of the shape of the population distribution, the sampling distribution of the sample proportion (or mean) will approach a normal distribution as the sample size becomes large enough.
• This theorem justifies using a normal approximation for the sampling distribution in our example, even though the population proportion is expressed in distinct categories (male and female).
• One essential implication of CLT is that it enables us to use the normal distribution for inference about the population, particularly when discussing probabilities and confidence intervals.

In sum, CLT is the backbone that allows us to use simple, standard statistical methods to explore complex data sets, ensuring analyses are both practical and reliable.