A cell phone service provider has selected a random sample of 20 of its customers in an effort to estimate the mean number of minutes used per day. The results of the sample included a sample mean of 34.5 minutes and a sample standard deviation equal to 11.5 minutes. Based on this information, and using a 95 percent confidence level:

Question

BlackTom AI · Accepted Answer

To begin, let's identify the key pieces of information: the sample size is 20, so degrees of freedom for the t-distribution would be 19, and we are using a 95% confidence level.
Option A states the critical value is t = 2.093. With df = 19 and a two-sided 95% confidence interval, the appropriate critical value is indeed t_{0.025,19} which is approximately 2.093, so this matches the standard t-table value.
Option B suggests z = 1.645. This value corresponds to a one-sided 90% or a two-sided 80% interval, not a 95% two-sided interval, and uses the normal distribution rather than the t distribution, which is not appropriate here given the sample size.
Option C suggests z = 1.96. This is the standard critical value for a 95% confidence interval when using the normal distribution (z), but since the population standard deviation is unknown and the sample standard deviation is used with a small sample size (n = 20), the t distribution should be used instead of z, making this option inaccurate in this context.
Option D claims the critical value can't be determined without the margin of error. In fact, the critical value depends only on the chosen confidence level and the sample size (through df = n - 1); the margin of error is a subsequent calculation, not a prerequisite for identifying the correct critical value. Therefore this statement is misleading.
Putting it all together, the only choice that correctly identifies the appropriate critical value for this scenario is the first option, t = 2.093, given the 95% confidence level and 19 degrees of freedom.

OSC 2030-009 Fall 2025 SUBMIT: Graded Homework Module 7 (Chap 8)

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