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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.4.36
Chapter 5, Problem 5.4.36

Milk Containers A machine is set to fill milk containers with a mean of 64 ounces and a standard deviation of 0.11 ounce. A random sample of 40 containers has a mean of 64.05 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain.

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The machine is functioning correctly, and the sample mean is not unusual (μ = 64). H₁: The machine is not functioning correctly, and the sample mean is unusual (μ ≠ 64).
Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is SE = σ / √n, where σ is the population standard deviation (0.11 ounces) and n is the sample size (40).
Step 3: Compute the z-score to determine how many standard errors the sample mean (64.05 ounces) is away from the population mean (64 ounces). The formula for the z-score is z = (x̄ - μ) / SE, where x̄ is the sample mean, μ is the population mean, and SE is the standard error calculated in Step 2.
Step 4: Compare the calculated z-score to the critical z-value for a two-tailed test at a chosen significance level (e.g., α = 0.05). For α = 0.05, the critical z-values are approximately ±1.96. If the z-score falls outside this range, the sample mean is considered unusual.
Step 5: Based on the comparison in Step 4, decide whether to reject the null hypothesis. If the z-score is within the critical range, the machine does not need to be reset. If the z-score is outside the critical range, the machine needs to be reset.

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Key Concepts

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

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. In this case, the mean of the sample of 40 containers will follow a normal distribution due to the Central Limit Theorem, which states that the distribution of the sample mean will approximate a normal distribution as the sample size increases.
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Standard Error

Standard error is the standard deviation of the sampling distribution of a statistic, commonly the sample mean. It quantifies how much the sample mean is expected to vary from the true population mean. For this scenario, the standard error can be calculated using the formula: standard deviation divided by the square root of the sample size, which helps determine if the sample mean is significantly different from the population mean.
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Calculating Standard Deviation

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this context, we would set up a null hypothesis stating that the machine is functioning correctly (mean = 64 ounces) and an alternative hypothesis suggesting that it is not. By calculating the z-score for the sample mean and comparing it to a critical value, we can determine if the observed mean is unusual enough to warrant resetting the machine.
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Step 1: Write Hypotheses
Related Practice
Textbook Question

Repeat Exercise 26 for samples of size 72 and 108. What happens to the mean and the standard deviation of the distribution of sample means as the sample size increases?

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x<109)


a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)

Textbook Question

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.


Textbook Question

Finding Area

In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.


To the right of z= -0.355

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 ≥ 110)

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 = 1275, sigma =6, n = 1000