Paint Cans A machine is set to fill paint cans with a mean of 128 ounces and a standard deviation of 0.2 ounce. A random sample of 40 cans has a mean of 127.9 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain.

Verified step by step guidance

1. Step 1: Identify the problem as a hypothesis test for the sample mean. The null hypothesis (H₀) is that the machine is functioning correctly, with a mean of 128 ounces. The alternative hypothesis (Hₐ) is that the machine is not functioning correctly, meaning the sample mean is significantly different from 128 ounces.
2. Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is: $\mathrm{SE}=\frac{\sigma }{\sqrt{n}}$, where σ is the population standard deviation (0.2 ounces) and n is the sample size (40).
3. Step 3: Compute the z-score to determine how many standard errors the sample mean (127.9 ounces) is away from the population mean (128 ounces). The formula for the z-score is: $z=\frac{\mathrm{x̄}-\mu }{\mathrm{SE}}$, where x̄ is the sample mean, μ is the population mean, and SE is the standard error calculated in Step 2.
4. Step 4: Compare the calculated z-score to the critical z-value for a given significance level (commonly α = 0.05 for a two-tailed test). The critical z-values for α = 0.05 are approximately ±1.96. If the calculated z-score falls outside this range, the sample mean is considered unusual.
5. Step 5: Based on the comparison in Step 4, decide whether to reject the null hypothesis. If the z-score is unusual (outside ±1.96), conclude that the machine needs to be reset. Otherwise, conclude that the machine does not need to be reset.

Key Concepts

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 cans 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.

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 of 127.9 ounces is significantly different from the population mean of 128 ounces.

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 = 128 ounces) and an alternative hypothesis suggesting 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.