The amount of time spent by Australian children playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours.  Find the probability that if four Australian children are randomly selected, their average number of hours spent playing sport is more than 5 hours per day.

To solve this problem, we need to use the concept of the sampling distribution of the sample mean.

Step 1: Identify the parameters of the population distribution The problem tells us that the amount of time spent by Australian children playing sports per day is normally distributed with a mean (μ) of 4 hours and a standard deviation (σ) of 1.25 hours.

Step 2: Identify the parameters of the sampling distribution The mean of the sampling distribution (μx̄) is equal to the population mean, which is 4 hours. The standard deviation of the sampling distribution (σx̄), also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size (n). In this case, n = 4, so σx̄ = σ/√n = 1.25/√4 = 0.625 hours.

Step 3: Standardize the desired sample mean We want to find the probability that the sample mean is more than 5 hours. To do this, we first convert 5 hours to a z-score, which is a measure of how many standard errors the value is away from the mean of the sampling distribution. The z-score is calculated as follows: z = (x̄ - μx̄)/σx̄ = (5 - 4)/0.625 = 1.6.

Step 4: Find the probability associated with the z-score The z-score tells us that 5 hours is 1.6 standard errors above the mean. Because we want to find the probability that the sample mean is more than 5 hours, we need to find the area to the right of this z-score on the standard normal distribution. Looking up 1.6 on a standard normal (Z) table or using a Z-table calculator, we find that the area to the left is approximately 0.9452. However, since we are interested in the area to the right, we subtract this value from 1 (because the total area under the curve is 1), giving us 1 - 0.9452 = 0.0548.

So, the probability that the average number of hours spent playing sport by four randomly selected Australian children is more than 5 hours per day is approximately 0.0548, or 5.48%.