Hypothesis Test with Known σ


a. How do the results from Example 1 in this section change if σ is known to be 1.99240984 g? Does the knowledge of σ have much of an effect on the results of this hypothesis test?

Verified step by step guidance

1. Step 1: Begin by identifying the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis typically states that there is no effect or no difference, while the alternative hypothesis represents the claim being tested.
2. Step 2: Since σ (the population standard deviation) is known, use the z-test formula for hypothesis testing. The formula is: $z=(\mathrm{x̄}-\mu )\sigma /\sqrt{n}$, where x̄ is the sample mean, μ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.
3. Step 3: Calculate the z-score using the given values for x̄, μ, σ, and n. Plug these values into the formula to compute the test statistic.
4. Step 4: Determine the critical z-value(s) based on the significance level (α) and whether the test is one-tailed or two-tailed. Compare the calculated z-score to the critical z-value(s) to decide whether to reject or fail to reject the null hypothesis.
5. Step 5: Reflect on the impact of knowing σ. When σ is known, the z-test is more precise because it uses the actual population standard deviation rather than an estimate. This can slightly affect the results, especially if the sample size is small or the estimated standard deviation differs significantly from the true σ.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The outcome is assessed using a significance level, which indicates the probability of making a Type I error.

Standard Deviation (σ)

Standard deviation (σ) is a measure of the amount of variation or dispersion in a set of values. In hypothesis testing, knowing the population standard deviation allows for more accurate calculations of the test statistic and confidence intervals. When σ is known, it simplifies the process of determining the critical values and p-values in hypothesis tests.

Effect of Known σ on Hypothesis Testing

When the population standard deviation (σ) is known, it can significantly affect the results of a hypothesis test. Specifically, it allows the use of the Z-test, which is more powerful and provides more precise results compared to the t-test used when σ is unknown. This knowledge can lead to narrower confidence intervals and more reliable conclusions about the population parameter being tested.