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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 5 - Normal Probability DistributionsProblema 5.R.55b
Capitolo 5, Problema 5.R.55b

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (b) between 6000 and 6500 MMT CO2 eq. Compare your answers with those in Exercise 33.

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Step 1: Identify the key components of the problem. You are tasked with finding the probability that the mean amount of greenhouse gases for a random sample of 2 years is between 6000 and 6500 MMT CO2 eq. This involves using the sampling distribution of the sample mean.
Step 2: Recall the formula for the sampling distribution of the sample mean. The mean of the sampling distribution is the population mean (μ), and the standard deviation of the sampling distribution (σₓ̄) is given by: σx¯=σn, where σ is the population standard deviation and n is the sample size.
Step 3: Standardize the values 6000 and 6500 using the z-score formula: z=(x-μ)σx¯. This will convert the raw values into z-scores, which can then be used to find probabilities from the standard normal distribution.
Step 4: Use the z-scores obtained in Step 3 to find the cumulative probabilities from the standard normal distribution table or a statistical software. Subtract the cumulative probability corresponding to the lower z-score (6000) from the cumulative probability corresponding to the upper z-score (6500) to find the probability that the sample mean is between these two values.
Step 5: Compare the calculated probability with the results from Exercise 33. Interpret the findings by discussing how the sample size and the sampling distribution affect the probability, and whether the results align with expectations based on Exercise 33.

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Sampling Distribution

The sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the sample means vary and is crucial for understanding how likely it is to obtain a sample mean within a certain range, such as between 6000 and 6500 MMT CO2 eq.
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Sampling Distribution of Sample Proportion

Central Limit Theorem (CLT)

The Central Limit Theorem states that, regardless of the population's distribution, the distribution of the sample means will approach a normal distribution as the sample size increases. This theorem is essential for calculating probabilities related to sample means, especially when determining the likelihood of the mean amount of greenhouse gases falling within a specified range.
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Calculating the Mean

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring, often using statistical methods. In this context, it requires using the properties of the normal distribution to find the probability that the sample mean of greenhouse gases lies between 6000 and 6500 MMT CO2 eq, which can be computed using z-scores and standard normal distribution tables.
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Probability From Given Z-Scores - TI-84 (CE) Calculator
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