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.

75 girls.

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Step 1: Identify the problem type. This is a problem involving the binomial distribution, where we are determining whether the observed number of girls (75 out of 100 births) is significantly low, significantly high, or neither.

Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 100, and the probability of success (p) for a girl being born is typically assumed to be 0.5 (assuming equal likelihood of boys and girls).

Step 3: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution using the formulas: μ = n × p and σ = √(n × p × (1 - p)).

Step 4: Determine the range for significantly low and significantly high values. A common rule is that values more than 2 standard deviations below the mean (μ - 2σ) are significantly low, and values more than 2 standard deviations above the mean (μ + 2σ) are significantly high.

Step 5: Compare the observed value (75 girls) to the calculated thresholds for significantly low and significantly high values. Based on this comparison, classify the result as (a) significantly low, (b) significantly high, or (c) 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 result in either a girl or a boy. Understanding this distribution helps in determining what counts as significantly low or high based on expected outcomes.

Significance Levels

Significance levels are thresholds used to determine whether an observed result is statistically significant. In this case, we assess whether the number of girls (75) is significantly low or high compared to the expected number based on a 50% probability for each gender. Common significance levels, such as 0.05, help in making these determinations by indicating how extreme a result must be to be considered unusual.

Subjective Judgment

Subjective judgment refers to personal interpretation and assessment based on individual beliefs or experiences rather than strict statistical criteria. In this exercise, subjective judgment is used to evaluate whether the number of girls born (75) is perceived as significantly low, high, or neither, which can vary based on the evaluator's perspective and understanding of typical birth ratios.

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