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The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. Essentially, it tells us how many standard deviations an element is from the mean. The calculation for the Z-score is given by the formula: \( Z = \frac{X - \text{mean}}{\text{standard deviation}} \) In the context of our problem, the Z-score helps us understand how likely it is for a randomly chosen male passenger to fit through the airplane doorway. By comparing the passenger's height of 72 inches to the mean male height of 68.6 inches and a standard deviation of 2.8 inches, we compute the Z-score. Here's the step-by-step breakdown:1. Subtract the mean height (68.6) from the doorway height (72), which results in 3.4.2. Then, divide 3.4 by the standard deviation (2.8).3. The computed Z-score is approximately \( \frac{3.4}{2.8} \approx 1.21 \).A positive Z-score indicates that the value is above the mean, while a negative Z-score shows it is below the mean.

The normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric about the mean. It depicts that most observations cluster around the central peak and are flanked by tails that decrease symmetrically. This distribution is important in statistics for various analyses, including the Z-score calculation in our exercise. Some crucial properties of the normal distribution are:- The mean, median, and mode of the distribution are all equal.- It is completely defined by its mean and standard deviation.- About 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three.In our case, we use the normal distribution to find probabilities associated with heights of male passengers. The mean height of 68.6 inches and the standard deviation of 2.8 inches describe our sample of male passengers. This allows us to use the Z-score to find the cumulative probability up to a Z-score of 1.21, which corresponds to approximately 88.69%. This means there is an 88.69% chance that a randomly selected male passenger will fit through the doorway without bending.

Sample mean probability is a concept used to determine the probability that the average of a sample will be a particular value. It is derived from the central limit theorem, which states that the distribution of sample means approaches a normal distribution as the sample size grows, regardless of the population's distribution shape. For our exercise, we consider 100 male passengers (a sample) to find the probability that their mean height is less than 72 inches. We use the formula:\[ Z = \frac{\bar{X} - \text{mean}}{\frac{\text{standard deviation}}{\text{sqrt}(n)}} \]Where \( \bar{X} \) is the sample mean height, the mean is the population mean, the standard deviation is the standard deviation for the population, and \( n \) is the sample size.In our problem:1. Calculate the standard deviation of the sample mean: \( \frac{2.8}{\text{sqrt}(100)} = 0.28 \).2. Subtract the population mean (68.6) from the sample mean (72), giving 3.4.3. Divide 3.4 by 0.28 to get 12.14.With a Z-score of 12.14, the probability that the sample mean height is less than 72 inches is almost 1 or 100%. This means it is almost certain that the average height of the 100 men will be below the doorway height of 72 inches. This calculation is particularly useful when assessing the average height of larger groups, rather than individual cases.