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Consider a Situation in Which We Expect One-Third of the Observed

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Consider a situation in which we expect one-third of the observed values to be in each of three categories. We can use an ?2 goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 14, 19, and 27, what is the P-value range we would obtain for our test? (Use the table of critical values shown to answer this question.)
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χ2 critical values  for df=2 Sig. level  Value 0.055.990.0257.380.019.21\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}

Recognize and apply the paired t-test for assessing the effectiveness of interventions on paired data.
Distinguish between paired and independent samples and select the appropriate statistical test.
Interpret the results of statistical tests, including confidence intervals and hypothesis tests, in the context of the research question.
Design experiments and observational studies to compare two groups using paired and independent samples.

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