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u > 46

z=0.95405567
p= .1700277191
x= 46.96
n= 80

What are the null and alternate hypothesis?

7 answers

In hypothesis testing, we typically set up two competing hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\) or \(H_a\)).

In your example, you have the condition \(u > 46\), which suggests that you might be testing whether the population mean (\(u\)) is greater than 46. Based on this, we can formulate the null and alternative hypotheses as follows:

  • Null Hypothesis (\(H_0\)): The population mean \(u\) is less than or equal to 46. \[ H_0: u \leq 46 \]

  • Alternative Hypothesis (\(H_a\)): The population mean \(u\) is greater than 46. \[ H_a: u > 46 \]

This setup indicates that you are conducting a one-tailed test to see if the mean is significantly greater than 46.

What is the value of the test statistic? Enter the value to the full accuracy shown (do not round).

To calculate the test statistic in a one-sample Z-test for the mean, you can use the following formula:

\[ Z = \frac{\bar{x} - u_0}{\frac{\sigma}{\sqrt{n}}} \]

where:

  • \( \bar{x} \) is the sample mean
  • \( u_0 \) is the population mean under the null hypothesis
  • \( \sigma \) is the standard deviation of the population (if known) or the sample standard deviation (if population standard deviation is not known)
  • \( n \) is the sample size

From your data:

  • \( \bar{x} = 46.96 \)
  • \( u_0 = 46 \) (since we are testing against \( u \leq 46 \))
  • \( n = 80 \)

However, you've provided \( z = 0.95405567 \), which seems to be calculated already. To calculate the test statistic, we would also need the standard deviation or standard error, which is not provided here.

For the sake of illustrative calculation, if we assume \( z = 0.95405567 \) represents the calculated value from the above formula or if you have the standard deviation, you can compute it as follows:

To obtain the test statistic without the value of the standard deviation, we can't proceed with an exact calculation.

If you do have the standard deviation, you can plug in the numbers directly to calculate the test statistic. If, instead, you just need to use the given test statistic directly, you can state:

\[ Z = 0.95405567 \]

Please provide the standard deviation if you'd like to calculate it from first principles, or confirm if you'd like to use the \( z \) value you've presented.

Do you reject Hv0 at the a=0.05 level?

To determine whether to reject the null hypothesis (\(H_0\)) at the significance level \(\alpha = 0.05\), you need to compare the calculated test statistic with the critical value from the Z-distribution for a one-tailed test.

  1. Determine the critical Z value: For a one-tailed test at the \(\alpha = 0.05\) level, the critical Z value is approximately 1.645. This means that if your calculated Z value is greater than 1.645, you will reject the null hypothesis.

  2. Compare the test statistic to the critical value:

    • Your test statistic \(Z = 0.95405567\).
    • The critical value at \(\alpha = 0.05\) (one-tailed) is \(1.645\).

Since \(0.95405567 < 1.645\), we do not reject the null hypothesis.

Conclusion:

At the \(\alpha = 0.05\) significance level, you do not reject \(H_0\). This suggests that there is not enough evidence to conclude that the population mean is greater than 46 based on the data provided.

Do you reject Hv0 at the a=0.01 level?

To determine whether to reject the null hypothesis (\(H_0\)) at the significance level \(\alpha = 0.01\), we again need to compare the calculated test statistic with the critical Z value for a one-tailed test.

  1. Determine the critical Z value: For a one-tailed test at the \(\alpha = 0.01\) level, the critical Z value is approximately 2.33. This means that if your calculated Z value is greater than 2.33, you will reject the null hypothesis.

  2. Compare the test statistic to the critical value:

    • Your test statistic \(Z = 0.95405567\).
    • The critical value at \(\alpha = 0.01\) (one-tailed) is \(2.33\).

Since \(0.95405567 < 2.33\), we do not reject the null hypothesis.

Conclusion:

At the \(\alpha = 0.01\) significance level, you do not reject \(H_0\). This suggests that there is insufficient evidence to conclude that the population mean is greater than 46 based on the data provided at this more stringent significance level.