Questions: For the given data, (a) find the test statistic, (b) find the standardized test statistic, (c) decide whether the standardized test statistic is in the rejection region, and (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent. Claim: μ1<μ2, α=0.01. Sample statistics: x̄1=1230, n1=40, x̄2=1190, and n2=70. Population parameters: σ1=75 and σ2=105.

For the given data, (a) find the test statistic, (b) find the standardized test statistic, (c) decide whether the standardized test statistic is in the rejection region, and (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent. Claim: μ1<μ2, α=0.01. Sample statistics: x̄1=1230, n1=40, x̄2=1190, and n2=70. Population parameters: σ1=75 and σ2=105.
Transcript text: For the given data, (a) find the test statistic, (b) find the standardized test statistic, (c) decide whether the standardized test statistic is in the rejection region, and (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent. Claim: $\mu_{1}<\mu_{2}, \alpha=0.01$. Sample statistics: $\bar{x}_{1}=1230, n_{1}=40, \bar{x}_{2}=1190$, and $n_{2}=70$. Population parameters: $\sigma_{1}=75$ and $\sigma_{2}=105$.
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Solution

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Solution Steps

Step 1: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula:

\[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{75^2}{40} + \frac{105^2}{70}} = \sqrt{\frac{5625}{40} + \frac{11025}{70}} \approx 17.2663 \]

Step 2: Calculate the Test Statistic

The test statistic \( z \) is computed as follows:

\[ z = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{1230 - 1190}{17.2663} \approx 2.3167 \]

Step 3: Calculate the P-value

The P-value for a two-tailed test is given by:

\[ P = 2 \times (1 - Z(|z|)) \approx 0.0205 \]

Step 4: Determine the Rejection Region

For a left-tailed test with significance level \( \alpha = 0.01 \), the critical z-value is approximately \( -2.33 \). Since \( z \approx 2.3167 \) is greater than \( -2.33 \), it does not fall in the rejection region.

Final Answer

(a) The test statistic (Z) is: \( \boxed{2.3167} \)
(b) The standardized test statistic is: \( \boxed{2.3167} \)
(c) Is the standardized test statistic in the rejection region? \( \boxed{\text{False}} \)

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