More IQ Scores The population of IQ scores of adults is normally distributed. If we obtain a voluntary response sample of 5000 of those IQ scores, will a histogram of the sample be bell-shaped?

Verified step by step guidance

1. Understand the concept of a voluntary response sample: This type of sample is self-selected, meaning individuals choose to participate. It often leads to bias because those who respond may not represent the entire population.
2. Recall the properties of a normal distribution: A normal distribution is symmetric and bell-shaped, characterized by its mean and standard deviation.
3. Consider the Central Limit Theorem: This theorem states that the distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the population distribution. However, it applies to the distribution of sample means, not the distribution of individual sample values.
4. Evaluate the impact of sample bias: Since the sample is voluntary, it may not accurately reflect the population's normal distribution. This could affect the shape of the histogram, making it potentially not bell-shaped.
5. Conclude based on sample characteristics: Due to the voluntary nature of the sample, the histogram may not be bell-shaped even if the population distribution is normal. The sample may be skewed or have other biases that affect its shape.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean and standard deviation, where most of the data points cluster around the mean, and the probabilities for values taper off equally in both directions. Understanding this concept is crucial as it describes the population of IQ scores in the question.

Voluntary Response Sample

A voluntary response sample is a type of non-probability sampling where participants self-select into the sample. This method often leads to bias because individuals with strong opinions or interests are more likely to participate, which may not accurately represent the entire population. Recognizing this bias is essential to evaluate whether the sample's histogram will reflect the population's normal distribution.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where data is grouped into bins or intervals. The height of each bar reflects the frequency of data points within each interval. In the context of the question, understanding how a histogram can visually represent the distribution of IQ scores helps assess whether the sample's distribution is bell-shaped like the population's normal distribution.