Two populations have normal distributions. The first has...

Two populations have normal distributions. The first has population standard deviation 2 and the second has population standard deviation $3 .$ A random sample of 16 measurements from the first population had a sample mean of $20 .$ An independent random sample of 9 measurements from the second population had a sample mean of $19 .$ Test the claim that the population mean of the first population exceeds that of the second. Use a $5 \%$ level of significance. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value. (d) Find the $P$ -value of the sample test statistic. (e) Conclude the test (f) Interpret the results.

Two populations have normal distributions. The first has population standard deviation 2 and the second has population standard deviation $3 .$ A random sample of 16 measurements from the first population had a sample mean of $20 .$ An independent random sample of 9 measurements from the second population had a sample mean of $19 .$ Test the claim that the population mean of the first population exceeds that of the second. Use a $5 \%$ level of significance.
(a) Check Requirements What distribution does the sample test statistic follow? Explain.
(b) State the hypotheses.
(c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value.
(d) Find the $P$ -value of the sample test statistic.
(e) Conclude the test
(f) Interpret the results.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. In this context, we set up a null hypothesis, which usually represents no effect or the status quo (e.g., the first population mean is less than or equal to the second), and an alternative hypothesis, which represents the claim being tested (e.g., the first population mean exceeds the second). This method involves calculating a test statistic, finding the corresponding p-value, and comparing it against a chosen significance level to decide whether to reject or fail to reject the null hypothesis.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of the mean values obtained from multiple random samples drawn from a population. When the population is normally distributed, or when the sample size is large by the Central Limit Theorem, the distribution of these sample means will be approximately normal. Because the population standard deviations are known in this scenario, the sampling distributions of the individual sample means are normal, and the difference between two independent sample means is also normally distributed.

Z-Test for the Difference Between Means

The Z-test is used when comparing two means from independent samples, particularly when the population standard deviations are known. The test statistic is calculated by taking the difference between the sample means and then standardizing this difference using the combined standard error of the two means. The resulting Z-value follows a standard normal distribution, which allows one to compute probabilities and p-values from standard normal tables or software.

P-value

The p-value in hypothesis testing is the probability of obtaining a test statistic as extreme as or more extreme than the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a small p-value indicates strong evidence to reject the null hypothesis. The p-value is directly compared to the predetermined significance level to decide whether the results are statistically significant.

Level of Significance and Decision Rule

The level of significance, commonly denoted as alpha (?), is the threshold set by the researcher (such as 5%) for deciding whether to reject the null hypothesis. If the p-value is less than or equal to the significance level, the null hypothesis is rejected in favor of the alternative hypothesis. This framework helps control the Type I error rate, which is the probability of falsely rejecting a true null hypothesis.

Interpretation of Statistical Results

Interpreting statistical results involves not just the mechanical decision of rejecting or failing to reject the null hypothesis, but also understanding the practical implications in the context of the problem. This includes explaining what the rejection (or failure to reject) means with respect to the original claim, and considering if the observed differences are practically significant or if they might have occurred by chance.

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