Final answer:
To find the standardized statistic for a sample proportion of 0.45, we calculate the estimated population proportion and margin of error from the confidence interval, determine the standard error, and then compute the z-score using these values.
Step-by-step explanation:
To calculate the standardized statistic, also known as the z-score, for a sample proportion of 0.45, we need to use the formula z = (p' - p) / SE, where p' is the sample proportion, p is the hypothesized population proportion, and SE is the standard error of the proportion. Given that we do not have the values for the hypothesized population proportion or the standard error directly, we need to deduce them from the confidence interval provided. Since we have a 95 percent confidence interval (0.113 to 0.439) for the population proportion, we can estimate the population proportion to be the midpoint of this interval, and the margin of error can be calculated as the difference between the upper bound and the estimated population proportion.
The estimated population proportion p is the midpoint of the confidence interval: p = (0.113 + 0.439) / 2 = 0.276. The margin of error (ME) is the difference between the upper bound of the interval and the estimated population proportion: ME = 0.439 - 0.276. Now we need to calculate the standard error (SE), which is the ME divided by the z-value that corresponds to the 95 percent confidence level (typically 1.96 for two-tailed tests): SE = ME / 1.96. Once we have the SE, we can then calculate the z-score using the formula mentioned earlier.
After calculating the standard error, the z-score for the sample proportion is calculated as z = (0.45 - 0.276) / SE. This value will then be rounded to three decimal places as requested.