A random sample is obtained from a population with variance σ^2=625, and the sample mean is comp

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A random sample is obtained from a population with variance...

A random sample is obtained from a population with variance $\sigma^{2}=625$, and the sample mean is computed. Test the null hypothesis $H_{0}: \mu=100$ versus the alternative hypothesis $H_{1}: \mu>100$ with $\alpha=0.05$. Compute the critical value $\bar{x}_{c}$ and state your decision rule for the following options. a. Sample size $n=25$ b. Sample size $n=16$ c. Sample size $n=44$ d. Sample size $n=32$

A random sample is obtained from a population with variance $\sigma^{2}=625$, and the sample mean is computed. Test the null hypothesis $H_{0}: \mu=100$ versus the alternative hypothesis $H_{1}: \mu>100$ with $\alpha=0.05$. Compute the critical value $\bar{x}_{c}$ and state your decision rule for the following options.
a. Sample size $n=25$
b. Sample size $n=16$
c. Sample size $n=44$
d. Sample size $n=32$

Key Concepts

Critical Value

The critical value corresponds to the threshold in the distribution of the test statistic beyond which the null hypothesis is rejected. For a one-tailed test at the 0.05 significance level using the standard normal distribution, the critical z-value is typically 1.645. This value is adjusted into the context of the sample mean by multiplying it by the standard error and adding it to the hypothesized mean, yielding the critical value of the sample mean (x?c).

Decision Rule

The decision rule in hypothesis testing stipulates that if the computed test statistic exceeds the critical value (or if the sample mean is greater than the calculated critical value), then the null hypothesis is rejected in favour of the alternative hypothesis. Otherwise, the null hypothesis is not rejected. This rule helps in making a clear decision based on the calculated evidence from the sample data.

Z-test Statistic

The Z-test is used when performing hypothesis tests for a population mean if the population variance is known and/or due to the application of the Central Limit Theorem for sample means. The test statistic is calculated by taking the difference between the sample mean and the hypothesized population mean, and then standardizing this difference by dividing by the standard error. This transforms the statistic into a standard normal variable (Z), which can then be compared against critical values.

Significance Level

The significance level (denoted ?) is the threshold probability used to decide whether to reject the null hypothesis. It quantifies the risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true. In this context, an ? of 0.05 means there is a 5% risk of wrongly rejecting H?.

Standard Error of the Sample Mean

The standard error measures how far the sample mean is likely to be from the true population mean. It is calculated by taking the square root of the population variance divided by the sample size (?/?n). A larger sample size decreases the standard error, leading to a more precise estimate of the population mean.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents a statement of no effect or no difference—here that the population mean is equal to a specified value. The alternative hypothesis (H?) represents the claim to be tested—in this case, that the true mean is greater than the specified value. These hypotheses set the framework for deciding whether there is sufficient evidence in the sample data to support a deviation from the null assumption.

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