Question

The lengths of text messages have an unknown distribution with mean 32 and standard deviation 4 characters. A sample, with size n=49, is randomly drawn from the population and the sum is taken. What is the probability that the sum is between 1585 and 1602 characters?

Answer 1

To find the probability that the sum of the text message lengths is between 1585 and 1602 characters, we use the Central Limit Theorem. The probability is approximately 0.1696, or 16.96%

Step-by-step explanation:

To find the probability that the sum of the text message lengths is between 1585 and 1602 characters, we need to use the Central Limit Theorem. This theorem states that for a large sample size, the sum of a random variable will be approximately normally distributed.

First, we need to calculate the mean and standard deviation of the sum of the text message lengths. The mean of the sum is equal to the mean of the individual text message lengths multiplied by the sample size (n), in this case 49. So the mean is 32 x 49 = 1568.

The standard deviation of the sum is equal to the standard deviation of the individual text message lengths multiplied by the square root of the sample size. So the standard deviation is 4 x sqrt(49) = 4 x 7 = 28.

Now, we can standardize the values 1585 and 1602 using the z-score formula, z = (x - mean) / standard deviation. The z-score for 1585 is (1585 - 1568) / 28 = 0.6071. The z-score for 1602 is (1602 - 1568) / 28 = 1.2143.

Finally, we can use a standard normal distribution table or a calculator to find the probability between these two z-scores. Using the table or calculator, we find that P(0.6071 < z < 1.2143) is approximately 0.1696. Therefore, the probability that the sum of the text message lengths is between 1585 and 1602 characters is approximately 0.1696, or 16.96%.