In Exercises 9–12, assume that 100 births are randomly selected. ...

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

2:34 minutes

Problem 4.1.9

Textbook Question

In Exercises 9–12, assume that 100 births are randomly selected. Use subjective judgment to describe the given number of girls as (a) significantly low, (b) significantly high, or (c) neither significantly low nor significantly high.

53 girls.

Verified step by step guidance

Step 1: Define the problem context. We are tasked with determining whether the observed number of 53 girls out of 100 births is significantly low, significantly high, or neither. This involves comparing the observed value to the expected distribution under the assumption of a fair 50/50 chance of having a girl or boy at each birth.

Step 2: Identify the statistical distribution. Since the births are independent and there are two possible outcomes (girl or boy), the number of girls in 100 births follows a binomial distribution with parameters n = 100 (number of trials) and p = 0.5 (probability of a girl).

Step 3: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = n × p, and the standard deviation is given by σ = √(n × p × (1 - p)). Substitute n = 100 and p = 0.5 into these formulas.

Step 4: Determine the range for significantly low and significantly high values. A common rule is to consider values more than 2 standard deviations away from the mean as significant. Calculate the lower threshold as μ - 2σ and the upper threshold as μ + 2σ.

Step 5: Compare the observed value (53 girls) to the thresholds. If the value is below the lower threshold, it is significantly low. If it is above the upper threshold, it is significantly high. If it falls within the range, it is neither significantly low nor significantly high.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In the context of births, we can model the number of girls born using a binomial distribution, where each birth can be considered a trial with two outcomes: girl or boy. Understanding this distribution helps in determining what counts as significantly low or high based on expected probabilities.

Significance Level

The significance level is a threshold used to determine whether a result is statistically significant. In this context, it helps to assess whether the observed number of girls (53) is significantly low or high compared to what would be expected under normal circumstances (typically around 50 girls in 100 births). A common significance level is 0.05, which indicates that results beyond this threshold are considered statistically significant.

Subjective Judgment

Subjective judgment refers to the process of making decisions based on personal opinions, interpretations, or feelings rather than objective data. In this exercise, subjective judgment is used to evaluate whether the number of girls (53) is perceived as significantly low, high, or neither, which can vary based on individual perspectives and the context of the situation.

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In Exercises 9–12, assume that 100 births are randomly selected. Use subjective judgment to describe the given number of girls as (a) significantly low, (b) significantly high, or (c) neither significantly low nor significantly high.53 girls.