Moving to another question will save this response. Question 8 Question 8 0.5 points In testing the hypotheses \( H_{0}: \mu=50 \mathrm{vs} \). \( H_{1}: \mu=50 \), the following information is known: \( n=64, \bar{x}=53.5 \), and \( \sigma=10 \). The standardized test statistic \( z \) equals: a. -2.80 b. 1.96 c. 2.80 d. -1.96 A Moving to another question will save this response. Question 8 of 15

To find the standardized test statistic \( z \), you can use the formula: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \] Substituting the values given: \( \bar{x} = 53.5 \), \( \mu = 50 \), \( \sigma = 10 \), and \( n = 64 \): \[ z = \frac{53.5 - 50}{\frac{10}{\sqrt{64}}} = \frac{3.5}{\frac{10}{8}} = \frac{3.5}{1.25} = 2.8 \] So, the answer is c. 2.80. The z-score tells us how many standard deviations the sample mean is from the population mean under the null hypothesis. A high z-score indicates that the sample mean is significantly different from the hypothesized value, helping researchers draw conclusions about their data. In the real world, this statistical calculation is crucial in numerous fields, such as psychology or sociology, where researchers frequently test hypotheses about population means. With enough data, the z-test helps them understand whether observed differences truly reflect the underlying populations or are just due to sampling variability.