Dole Pineapple Inc. is concerned that the 16 -ounce can of...

Dole Pineapple Inc. is concerned that the 16 -ounce can of sliced pineapple is being overfilled. Assume the standard deviation of the process is .03 ounce. The quality control department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the $5 \%$ level of significance, can we conclude that the mean weight is greater than 16 ounces? Determine the $p$ -value.

Key Concepts

Significance Level

The significance level, often denoted by ?, is the threshold for deciding whether an observed effect is statistically significant. It is the probability of rejecting the null hypothesis when it is actually true (a Type I error). A commonly used significance level is 5%, meaning that if the p-value is less than 0.05, the null hypothesis is rejected.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating a null hypothesis (usually representing a default or no?change situation) and an alternative hypothesis (representing a claim or effect to be tested), then using sample evidence to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis typically states that there is no effect or no difference, while the alternative hypothesis represents the claim that is being tested. In the context of testing whether the mean is greater than a specified value, the alternative hypothesis is one-sided, indicating that the mean is greater than the stated value.

Z-Test Statistic

The Z-test statistic is used when the population standard deviation is known and the sample size is sufficiently large. It measures how many standard errors the sample mean is away from the hypothesized population mean, using the formula Z = (observed mean – hypothesized mean) / (standard error).

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it quantifies the variability expected in the sample mean from one sample to another.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. In a one-sided test, it represents the area in the tail of the distribution beyond the calculated test statistic. A small p-value indicates that the observed data is unlikely under the null hypothesis, providing evidence against it.

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