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The Z-score is a measure that describes a value's position relative to the mean of a group of values. In simpler terms, it tells us how many standard deviations a specific value (X) is from the mean (µ). The formula for calculating the Z-score is: \[ Z = \frac{X - \text{µ}}{\text{σ}} \] The mean (µ) is the average value, and the standard deviation (σ) indicates how spread out the values are. In our example, we used the weights of women to calculate the Z-scores for 140 lb and 211 lb. For 140 lb, the Z-score is calculated as follows: \[ Z = \frac{140 - 171.1}{46.1} = -0.675 \] This means that 140 lb is -0.675 standard deviations from the mean. Similarly, for 211 lb, the Z-score is: \[ Z = \frac{211 - 171.1}{46.1} = 0.866 \] This indicates that 211 lb is 0.866 standard deviations above the mean.

Calculating the percentage is essential to understand the proportion of data within a certain range. In our exercise, we needed to find out what percentage of women fall between the weights of 140 lb and 211 lb. After calculating the Z-scores, we used a Z-score table or calculator to find the corresponding probabilities. For a Z-score of -0.675, the probability is approximately 0.250. For a Z-score of 0.866, the probability is approximately 0.807. To find the percentage of women within the weight range, we subtract the lower probability from the upper probability: \[ 0.807 - 0.250 = 0.557 \] We then convert this value to a percentage by multiplying by 100: \[ 0.557 \times 100 = 55.7\text{\textpercent} \] Therefore, 55.7% of women fall within the given weight limits.

Standard deviation (σ) is a measure of the amount of variation or dispersion in a set of values. It tells us how much the values in a dataset deviate from the mean. A low standard deviation indicates that the values are close to the mean, while a high standard deviation suggests that the values are spread out over a wider range. In our exercise, the standard deviation of women's weights is 46.1 lb. This means that on average, women's weights differ from the mean weight (171.1 lb) by 46.1 lb. Standard deviation helps in understanding the distribution of data, which is crucial for making accurate calculations and predictions.

The mean (µ) is the average value of a dataset and is calculated as the sum of all values divided by the number of values. It serves as a measure of the central tendency of the data. In our exercise, the mean weight of women is given as 171.1 lb. This value is central to our calculations and helps in determining the Z-scores. The mean provides a reference point for understanding how individual values compare to the overall dataset. To calculate the mean manually, you add up all the weights and divide by the number of women. For instance, if the weights are 140 lb, 150 lb, and 170 lb: \[ \text{Mean} = \frac{140 + 150 + 170}{3} = 153.33 \text{ lb} \] Knowing the mean is essential for further statistical calculations, such as finding the standard deviation and Z-scores.