In Exercises 27–32, the random variable x is normally distributed...

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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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1:22 minutes

Problem 5.RE.28

Textbook Question

In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(x < 55)

Verified step by step guidance

Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 74 and a standard deviation (σ) of 8. The goal is to find the probability P(x < 55).

Step 2: Standardize the value of x = 55 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (55 - 74) / 8.

Step 3: Simplify the z-score formula to calculate the z-value. This will give you the standardized value corresponding to x = 55.

Step 4: Use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to the calculated z-value. This cumulative probability represents P(x < 55).

Step 5: Interpret the result. The cumulative probability obtained from the z-table or software is the probability that the random variable x is less than 55 in the given normal distribution.

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Key Concepts

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

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (mu) and standard deviation (sigma). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean, and about 95% falls within two standard deviations. This distribution is fundamental in statistics for modeling real-world phenomena.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores allow for the comparison of scores from different normal distributions and are essential for finding probabilities in a standard normal distribution table.

Probability Calculation

Probability calculation in the context of a normal distribution involves determining the likelihood of a random variable falling within a certain range. This is often done using Z-scores to convert the values into a standard normal distribution, where probabilities can be easily found using statistical tables or software. For the question at hand, calculating P(x < 55) requires finding the Z-score for x = 55 and then using it to find the corresponding probability.

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In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.P(x < 55)