Final answer:
To calculate the probability of a sample mean falling between 19.5 and 20.5, the standard error of the mean for a sample of 144 should be determined, and Z-scores for 19.5 and 20.5 calculated. By referencing a Z-table, the probabilities corresponding to these Z-scores can be found and subtracted to give the required probability. However, the provided data does not match the scenario described in the question, suggesting it might be incomplete or incorrect.
Step-by-step explanation:
The question asks for the probability that a sample mean falls within a specific range, given that the population follows a normal distribution. The population mean (μ) is 36.9 years, and the population standard deviation (σ) is 13.9 years, as indicated by the provided information about smartphone users' ages. When working with the sample mean and a sample size of 144, the standard error of the mean (SEM) becomes relevant. The SEM is calculated by dividing the population standard deviation by the square root of the sample size.
To find the probability in question, one would standardize the range using the Z-score formula, which involves subtracting the population mean from the sample mean and then dividing by the SEM. Once the Z-scores for 19.5 and 20.5 are calculated, the cumulative distribution function (CDF) values from the Z-table will give the probabilities for values less than or equal to these Z-scores. Finally, the probability that the sample mean is between 19.5 and 20.5 would be the difference between these two probabilities.
The direct calculation described in this answer requires access to a Z-table or a statistical calculator, which would give a more precise answer, especially important as rounding can slightly alter results. To provide an exact probability, these calculations need to be performed with the given parameters.