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A test is made of Hv0 : u=18 versus Hv1: u does not = 18. A sample of size n = 45 is drawn, and x=16 . The population standard deviation is o=6.

(a) Compute the value of the test statistic z.
(b) Is Hv0 rejected at the a = 0.01 level?
(c) Is Hv0 rejected at the a= 0.05 level?

3 answers

To solve this hypothesis testing problem, we can follow these steps.

(a) Compute the value of the test statistic z.

The formula for the z-test statistic when testing a population mean is given by:

\[ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \]

where:

  • \(\bar{x}\) is the sample mean,
  • \(\mu_0\) is the population mean under the null hypothesis,
  • \(\sigma\) is the population standard deviation, and
  • \(n\) is the sample size.

Given:

  • \(\bar{x} = 16\)
  • \(\mu_0 = 18\)
  • \(\sigma = 6\)
  • \(n = 45\)

Plugging in the values:

\[ z = \frac{16 - 18}{\frac{6}{\sqrt{45}}} \]

Calculating the denominator:

\[ \frac{6}{\sqrt{45}} \approx \frac{6}{6.7082} \approx 0.8944 \]

Now substituting:

\[ z = \frac{-2}{0.8944} \approx -2.2361 \]

Therefore,

\[ z \approx -2.24 \]

(b) Is \(H_0\) rejected at the \(\alpha = 0.01\) level?

For a two-tailed test at \(\alpha = 0.01\), we need to find the critical z-values. The critical values are:

\[ \pm z_{\alpha/2} = \pm z_{0.005} \]

From the standard normal distribution table, \(z_{0.005} \approx 2.576\).

Thus, our rejection regions are:

  • \(z < -2.576\)
  • \(z > 2.576\)

Since our calculated \(z \approx -2.24\), which does not fall into the rejection region (\(-2.576 < -2.24 < 2.576\)), we do not reject \(H_0\) at the \(\alpha = 0.01\) level.

(c) Is \(H_0\) rejected at the \(\alpha = 0.05\) level?

For a two-tailed test at \(\alpha = 0.05\), the critical z-values are:

\[ \pm z_{\alpha/2} = \pm z_{0.025} \]

From the standard normal distribution table, \(z_{0.025} \approx 1.96\).

Thus, our rejection regions are:

  • \(z < -1.96\)
  • \(z > 1.96\)

Since our calculated \(z \approx -2.24\), which falls into the rejection region (\(-2.24 < -1.96\)), we reject \(H_0\) at the \(\alpha = 0.05\) level.

Summary:

(a) The value of the test statistic \(z \approx -2.24\).

(b) \(H_0\) is not rejected at \(\alpha = 0.01\).

(c) \(H_0\) is rejected at \(\alpha = 0.05\).

Choose the correct type of hypothesis test. Then find the critical values for a=0.01 and a=0.05. Round your answers to three decimal places, if necessary.
Is Hv1 right-tailed, left-tailed or two-tailed? What are the critical values for a=0.01? the critical values for a=0.05?

In your original hypothesis test, you stated the null hypothesis \(H_0: \mu = 18\) versus the alternative hypothesis \(H_1: \mu \neq 18\). Since the alternative hypothesis is testing for a mean not equal to a specified value, it indicates a two-tailed test.

Type of Hypothesis Test:

  • Type: Two-tailed test

Critical Values:

For a two-tailed test, the critical values will be determined based on the specified significance levels, \(\alpha\).

  1. For \(\alpha = 0.01\):

    • We divide the significance level by 2 for a two-tailed test: \[ \alpha/2 = 0.005 \]
    • Using the standard normal (z) distribution, the critical values at this level are: \[ z_{0.005} \approx \pm 2.576 \]
    • So, the critical values for \(\alpha = 0.01\) are: \[ -2.576 \quad \text{and} \quad 2.576 \]

  2. For \(\alpha = 0.05\):

    • Similarly, divide the significance level by 2: \[ \alpha/2 = 0.025 \]
    • From the standard normal distribution, the critical values at this level are: \[ z_{0.025} \approx \pm 1.96 \]
    • Therefore, the critical values for \(\alpha = 0.05\) are: \[ -1.96 \quad \text{and} \quad 1.96 \]

Summary:

  • Type of Test: Two-tailed
  • Critical Values for \(\alpha = 0.01\): \(-2.576\) and \(2.576\)
  • Critical Values for \(\alpha = 0.05\): \(-1.96\) and \(1.96\)