Consider the following hypothesis test: Ha: μ≤50 Ha: μ>50 A sample of 60 is used and

step 1 Hello, so here we're given that n is equal to 60. Sample mean X bar is 52 .5, and the population

Consider the following hypothesis test: Ha: μ≤50 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_a: \mu \leq 50 \\ & H_{\mathrm{a}}: \mu>50 \end{aligned} $$ A sample of 60 is used and the population standard deviation is 8 . Use the critical value approach to state your conclusion for each of the following sample results. Use $a=.05$. a. $\tilde{x}=52.5$ b. $\bar{x}=51$ c. $\overline{\boldsymbol{x}}=51.8$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_a: \mu \leq 50 \\
& H_{\mathrm{a}}: \mu>50
\end{aligned}
$$
A sample of 60 is used and the population standard deviation is 8 . Use the critical value approach to state your conclusion for each of the following sample results. Use $a=.05$.
a. $\tilde{x}=52.5$
b. $\bar{x}=51$
c. $\overline{\boldsymbol{x}}=51.8$

Key Concepts

Significance Level and Rejection Region

The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true, usually set at 0.05 or 5%. This defines the rejection region for the test statistic; if the computed value is in this region, the evidence is strong enough to reject the null hypothesis.

Critical Value Approach

The critical value approach involves comparing the calculated test statistic to a predetermined cutoff point (the critical value) derived from the significance level of the test. If the test statistic falls into the rejection region (beyond the critical value), the null hypothesis is rejected.

One-Sample Z-Test

A one-sample Z-test is used when the population standard deviation is known and the sample size is sufficiently large. It compares the sample mean to the hypothesized population mean by standardizing the difference, resulting in a Z-test statistic that can be compared to critical values from the standard normal distribution.

Null and Alternative Hypotheses

The null hypothesis (H0) generally represents the current or default state of affairs, while the alternative hypothesis (Ha) represents a new claim that is being tested. In a one-sided test, the alternative hypothesis specifies a direction of the effect, such as checking if a parameter is greater than a specific value.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to support a particular claim about a population parameter. It involves formulating a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis (the research claim), then using sample data to decide if the null hypothesis can be rejected.

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