Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.


COVID-19 Response Surveyors asked respondents to rate ten key aspects of their government’s response to the COVID-19 pandemic, including preparedness, communication, and material aid. A pandemic response score that ranged from 0 to 100 was calculated. The mean score for U.S. respondents was 50.6 with a standard deviation of 29.0. (Source: PLOS One)


b. What score represents the 61st percentile?

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the score that corresponds to the 61st percentile in a normal distribution. The mean (μ) is 50.6, and the standard deviation (σ) is 29.0. Percentiles represent the cumulative probability in a normal distribution.
2. Step 2: Convert the percentile (61%) into a z-score using the standard normal distribution table or a statistical tool. The z-score is a standardized value that corresponds to the cumulative probability of 0.61 in the standard normal distribution.
3. Step 3: Use the z-score formula to find the raw score (X) in the original distribution. The formula is: $X=\mu +z\times \sigma$, where μ is the mean, σ is the standard deviation, and z is the z-score.
4. Step 4: Substitute the values into the formula. Use μ = 50.6, σ = 29.0, and the z-score obtained from Step 2.
5. Step 5: Calculate the raw score (X) using the formula. This will give you the score that represents the 61st percentile in the given normal distribution.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding this concept is crucial for interpreting data in many fields, including statistics, as it helps in making inferences about population parameters based on sample statistics.

Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 61st percentile is the score below which 61% of the data points lie. This concept is essential for understanding the relative standing of a score within a distribution, allowing for comparisons between different scores and the overall distribution of data.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It indicates how many standard deviations an element is from the mean. Z-scores are particularly useful in normal distributions for determining percentiles, as they allow for the conversion of raw scores into a standardized form that can be easily compared across different datasets.