Question

A study indicates that 18- to 24-year olds spend a mean of 115 minutes watching video on their smanphones per month. Assume that the aenount of time watching video an a smartphone per month is norrmally distributed and that the standard deviafion is 20 minutes. Complete parts (a) through (d) below. a. What is the probability that an 18- to 24 year old spends less than 90 rainutes watching video or his or her smartphone per month? The probabilsty that an f8- to 24 year-old spends lass than 90 minutes watching video on his or her amatphone per montt is (Round to four decimal places as needed) b. What is the probability that an 18 - to 24 year-old spends between 90 and 145 minutos watching video on tis or her maraphone per moneh? The probability that an 13- to 24 yatar-old spends between 90 and 145 minutes sitchibg video on his or har mmartphone per month is (Round to four decinal places as needed.)

Answer 1

a. The probability that an 18- to 24-year-old spends less than 90 minutes watching video on their smartphone per month is approximately 0.3085.

b. The probability that an 18- to 24-year-old spends between 90 and 145 minutes watching video on their smartphone per month is approximately 0.3453.

Step-by-step explanation:

For part (a), to find the probability that an individual spends less than 90 minutes watching video on their smartphone, we use the standard normal distribution and z-score calculation. The formula for z-score is (X - μ) / σ, where X is the given value, μ is the mean, and σ is the standard deviation. In this case, z = (90 - 115) / 20, which equals -1.25. Using a standard normal distribution table or calculator, we find the corresponding probability, which is 0.1056. To get the probability less than 90 minutes, we add 0.5 (for the area to the left of the mean) to this value, resulting in 0.3085.

For part (b), we repeat the process for both 90 minutes and 145 minutes, finding their respective z-scores. The z-score for 90 minutes is -1.25 (as calculated in part a), and the z-score for 145 minutes is (145 - 115) / 20 = 1.5. Using the standard normal distribution table or calculator, we find the probabilities for these z-scores, which are 0.1056 and 0.0668, respectively. The probability between these two values is obtained by subtracting the smaller probability from the larger, resulting in 0.3453. This represents the likelihood that an 18- to 24-year-old spends between 90 and 145 minutes watching video on their smartphone per month.