Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.4.41

Chapter 5, Problem 5.4.41

Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.

Verified step by step guidance

Step 1: Identify the given information. The population proportion (p) is 0.63, the sample size (n) is 105, and we are interested in the probability that the sample proportion (p̂) is less than 0.55.

Step 2: Calculate the mean (μ) and standard deviation (σ) of the sampling distribution of the sample proportion. The mean is equal to the population proportion, μ = p = 0.63. The standard deviation is calculated using the formula: σ = sqrt((p * (1 - p)) / n).

Step 3: Standardize the sample proportion to find the z-score. Use the formula: z = (p̂ - μ) / σ, where p̂ is the sample proportion (0.55 in this case). Substitute the values of μ and σ from the previous step.

Step 4: Use the z-score to find the cumulative probability. Look up the z-score in the standard normal distribution table or use statistical software to find the probability that corresponds to the z-score.

Step 5: Interpret the result. The cumulative probability represents the likelihood that the sample proportion is less than 55%. This value can be interpreted in the context of the problem to understand how unusual it is for the sample proportion to be below 55% given the population proportion of 63%.

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Key Concepts

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

Sample Proportion

The sample proportion is the ratio of the number of individuals in a sample who exhibit a certain characteristic to the total number of individuals in that sample. In this context, it refers to the proportion of residents in favor of building a new high school among the 105 randomly selected residents. Understanding sample proportion is crucial for estimating population parameters and conducting hypothesis tests.

Normal Approximation to the Binomial Distribution

When dealing with proportions, especially in large samples, the sampling distribution of the sample proportion can be approximated by a normal distribution due to the Central Limit Theorem. This approximation is valid when both np and n(1-p) are greater than 5, where n is the sample size and p is the population proportion. This concept allows us to calculate probabilities related to sample proportions using the properties of the normal distribution.

Probability Interpretation

Probability interpretation involves understanding the likelihood of an event occurring within a given context. In this question, it requires interpreting the calculated probability that the sample proportion of residents in favor of the new school is less than 55%. This interpretation helps in making informed decisions based on statistical evidence and understanding the implications of the results in the context of community support for the school.

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Related Practice

Textbook Question

In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.

Mu = 150, sigma =25, n = 49

102

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Textbook Question

Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.

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Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 150)

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Textbook Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P91

127

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Textbook Question

True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

As the sample size increases, the mean of the distribution of sample means increases.

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