QuestionThe number of seconds $X$ after the minute that class ends is uniformly distributed between 0 and 60 . Round all answers to 4 decimal places where possible. a. What is the distribution of $X$ ? $X \sim U($ , $\square$ ) then the sampling distribution is b. Suppose that 38 classes are clocked. What is the distribution of $\bar{x}$ for this group of classes? $\bar{x} \sim N($ $\square$ $\square$ c. What is the probability that the average of 38 classes will end with the second hand between 27 and 31 seconds? $\square$

Studdy Solution
Calculate the probability that the sample mean is between 27 and 31 seconds.
Convert 27 and 31 to z-scores:
$$z_{27} = \frac{27 - 30}{2.8071} \approx -1.0684$$
$$z_{31} = \frac{31 - 30}{2.8071} \approx 0.3561$$
Use the standard normal distribution to find probabilities:
$$P(27 < \bar{x} < 31) = P(z_{27} < Z < z_{31})$$
Using standard normal distribution tables or a calculator:
$$P(Z < 0.3561) \approx 0.6388$$
$$P(Z < -1.0684) \approx 0.1423$$
Therefore:
$$P(27 < \bar{x} < 31) = 0.6388 - 0.1423 = 0.4965$$
The probability that the average of 38 classes will end with the second hand between 27 and 31 seconds is approximately:
$$\boxed{0.4965}$$