A random sample of 49 measurements from a population with...

A random sample of 49 measurements from a population with population standard deviation 3 had a sample mean of $10 .$ An independent random sample of 64 measurements from a second population with population standard deviation 4 had a sample mean of $12 .$ Test the claim that the population means are different. Use level of significance 0.01. (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.

A random sample of 49 measurements from a population with population standard deviation 3 had a sample mean of $10 .$ An independent random sample of 64 measurements from a second population with population standard deviation 4 had a sample mean of $12 .$ Test the claim that the population means are different. Use level of significance 0.01.
(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

Assumptions and Requirements

In order to perform a two-sample hypothesis test, certain conditions must be met. This includes having independent random samples from each population, known population standard deviations, and samples that are either from populations with normal distributions or with sufficiently large sample sizes so that the Central Limit Theorem applies. These assumptions ensure that the sampling distributions of the sample means, and consequently their difference, are well?approximated by a normal distribution.

Two-Sample Z-Test

The two-sample z-test is used when comparing the means of two independent populations when the population standard deviations are known. It involves calculating a test statistic based on the difference between the sample means, the known standard deviations, and the sizes of the samples. Because the test statistic follows a standard normal distribution under the null hypothesis, it allows for the assessment of whether the observed difference in sample means is statistically significant.

Hypothesis Formulation

In hypothesis testing, the null hypothesis typically posits that there is no difference between the population means, while the alternative hypothesis suggests that a difference exists. This framework sets the stage for deciding whether to reject the null hypothesis based on the observed data and the specified level of significance.

Sampling Distribution of the Difference in Means

The sampling distribution of the difference in sample means is key to performing the test. Under the assumption of independence and given known population variances, the difference between the sample means will also be normally distributed, with a mean equal to the difference of the population means and a standard deviation equal to the square root of the sum of the variances divided by their respective sample sizes.

Standard Error

The standard error measures the variability in the difference between the sample means. It is calculated using the known population standard deviations and the sample sizes, specifically as the square root of the sum of the squared standard deviations divided by the corresponding sample sizes. This quantity is critical because it scales the difference between sample means, allowing for the construction of the test statistic.

P-value and Statistical Significance

The P-value represents the probability of obtaining results as extreme as, or more extreme than, those observed, assuming that the null hypothesis is true. By comparing the P-value to the predetermined level of significance (alpha), one determines whether the observed difference could be due to chance, thereby guiding the decision to reject or fail to reject the null hypothesis.

Test Conclusion and Interpretation

The final step in hypothesis testing involves drawing a conclusion based on the P-value relative to the significance level. If the P-value is less than or equal to the significance level, the null hypothesis is rejected, indicating that there is statistically significant evidence of a difference between the population means. The interpretation should clearly state what the statistical decision means in the context of the populations being compared.

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