Bone Density Test. In Exercises 1–4, assume that scores on a bone...

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1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 3m
- Introduction to Confidence Intervals
  15m
- Confidence Intervals for Population Mean
  48m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample1h 1m
- Steps in Hypothesis Testing
  1h 1m
10. Hypothesis Testing for Two Samples2h 8m
11. Correlation48m
- Scatterplots & Intro to Correlation
  26m
- Correlation Coefficient
  21m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

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2:32 minutes

Problem 6.q.4

Textbook Question

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.

Bone Density For a randomly selected subject, find the probability of a bone density score between and 2.00.

Verified step by step guidance

Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. This is known as the standard normal distribution.

Step 2: Identify the range of interest for the bone density score, which is between 0 and 2.00. The goal is to find the probability that a randomly selected score falls within this range.

Step 3: Use the standard normal distribution table (Z-table) or a statistical software to find the cumulative probability for Z = 2.00. The Z-score formula is not needed here since the values are already standardized.

Step 4: Find the cumulative probability for Z = 0. Since the mean of the standard normal distribution is 0, the cumulative probability for Z = 0 is 0.5.

Step 5: Subtract the cumulative probability for Z = 0 from the cumulative probability for Z = 2.00 to find the probability of a score between 0 and 2.00. This difference represents the area under the curve between these two Z-scores.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing 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. In this context, the bone mineral density scores are assumed to follow a normal distribution with a mean of 0 and a standard deviation of 1.

Z-Scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. In this case, the Z-scores for the bone density scores will help determine the probability of a score falling between specific values, such as between 0 and 2.00.

Probability and Area Under the Curve

In statistics, the probability of a score falling within a certain range in a normal distribution can be found by calculating the area under the curve between those two points. This is typically done using Z-tables or statistical software. For the bone density test, finding the probability of scores between 0 and 2.00 involves determining the area under the normal curve between these Z-scores.

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Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.Bone Density For a randomly selected subject, find the probability of a bone density score between and 2.00.