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Answer :
The calculated value of the test statistic is -0.0869.
In hypothesis testing, the test statistic is a crucial measure used to assess the evidence against a null hypothesis. In this scenario, Bobs Wires is testing whether the machine is producing wire rolls different from the standard 100-ft rolls. The sample mean obtained from the 36 rolls is 99.9 feet.
To calculate the test statistic, we use the formula:
[tex]\[ \text{Test Statistic} = \frac{\text{Sample Mean} - \text{Population Mean}}{\text{Population Standard Deviation}/\sqrt{\text{Sample Size}}} \][/tex]
Plugging in the values:
Sample Mean = 99.9 ft,
Population Mean (expected) = 100 ft,
Population Standard Deviation = 6.9 ft,
Sample Size = 36.
[tex]\[ \text{Test Statistic} = \frac{99.9 - 100}{6.9/\sqrt{36}} = \frac{-0.1}{1.15} = -0.0869 \][/tex]
This calculated test statistic value of -2.03 represents how many standard deviations the sample mean is away from the population mean. In hypothesis testing, it helps determine whether the observed deviation is significant enough to reject the null hypothesis.
In this case, the negative value suggests the sample mean is lower than expected, indicating a potential deviation in wire length from the standard 100-ft rolls.
Hypothesis testing involves comparing sample data to a null hypothesis and interpreting the results based on the test statistic and p-value. The test statistic's magnitude indicates the significance of the difference between sample and expected values.
Negative values suggest the sample is lower than expected, while positive values suggest it is higher. Understanding the concept of test statistics is fundamental in making informed decisions about hypothesis testing outcomes.
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