Final Answer:
a) To approximate the probability that x will be within 0.3 of the population mean using the standard deviation, we need additional information such as the shape of the distribution or specific values for the mean and standard deviation.
b) To estimate the probability using the z-score, we can utilize the standard normal distribution table, finding the area under the curve corresponding to a z-score of ±0.3. This provides the probability that a randomly selected value falls within 0.3 standard deviations of the mean.
c) Applying the variance alone is insufficient for estimating probabilities directly. The variance gives a measure of the spread of data, but without the standard deviation, it doesn't provide a clear probability estimate.
d) Using the mode to approximate the probability is not a standard statistical practice. The mode represents the most frequently occurring value and doesn't inherently provide a measure of spread or probability within a specific range.
Step-by-step explanation:
When estimating the probability within a certain range of the population mean, the z-score is a valuable tool. The z-score standardizes a data point by expressing it in terms of standard deviations from the mean. By consulting a standard normal distribution table, we can find the probability associated with a z-score of ±0.3. For instance, if P(Z < -0.3) is 0.3821 and P(Z < 0.3) is 0.6179, the probability that x is within 0.3 of the mean is 0.6179 - 0.3821 = 0.2358.
Using variance alone is not practical for estimating probabilities, as it lacks the specific unit of measurement inherent in the standard deviation. The standard deviation, denoted as σ (sigma), provides a clearer picture of the spread of data around the mean. Employing the mode for probability estimation is unconventional and not recommended in standard statistical practice. The mode identifies the most frequent value but doesn't offer insights into the likelihood of values within a specified range. In conclusion, the z-score, derived from the standard deviation, is the preferred method for estimating the probability that x will be within 0.3 of the population mean.