6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:18 minutesProblem 6.6.15a
Textbook Question
Smartphones Based on an LG smartphone survey, assume that 51% of adults with smartphones use them in theaters. In a separate survey of 250 adults with smartphones, it is found that 109 use them in theaters.
a. If the 51% rate is correct, find the probability of getting 109 or fewer smartphone owners who use them in theaters.
Verified step by step guidance1
Step 1: Identify the type of probability distribution. Since we are dealing with a fixed number of trials (250 adults), two possible outcomes (use in theaters or not), and a constant probability of success (51%), this is a binomial distribution problem.
Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 250, the probability of success (p) is 0.51, and the number of successes (x) is 109.
Step 3: Convert the binomial distribution to a normal distribution for approximation. The mean (μ) and standard deviation (σ) of the binomial distribution are calculated as follows: μ = n * p and σ = sqrt(n * p * (1 - p)).
Step 4: Apply the continuity correction. Since we are finding the probability of getting 109 or fewer successes, adjust the value of x to 109.5 for the normal approximation.
Step 5: Standardize the value using the z-score formula: z = (x - μ) / σ. Then, use the standard normal distribution table or a statistical software to find the cumulative probability corresponding to the calculated z-score.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the success is defined as an adult using their smartphone in a theater. The parameters include the number of trials (n = 250) and the probability of success (p = 0.51).
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Normal Approximation to the Binomial
For large sample sizes, the binomial distribution can be approximated by a normal distribution. This is applicable when both np and n(1-p) are greater than 5. In this case, we can use the normal approximation to calculate the probability of observing 109 or fewer smartphone users in theaters, simplifying the calculations.
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Cumulative Probability
Cumulative probability refers to the probability of a random variable being less than or equal to a certain value. In this scenario, we need to calculate the cumulative probability of observing 109 or fewer users in theaters, which can be found using the normal distribution's cumulative distribution function (CDF) after applying the normal approximation.
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