The Theme park is facing a problem of determining the limits of the height that would include at least 90% of the children that attend the theme park. To solve this problem the theme park can apply the central limit theorem to establish the height limits for each age group. The central limit theorem states that the mean for a sufficiently large population will be normally distributed (Curwin & Slater, 2008). Therefore, given the standard deviation and the mean of each set, the theme park can construct confidence intervals for each group. To ensure at least 90% of the children attending the theme park meet the criteria, the relevant confidence interval is a 90% confidence interval for each age group calculated as follows.
The 90% confidence interval for five year olds ranges between 0.286 meters and 2.002 meters. The 90% confidence interval for twelve year olds ranges between 0.396 meters and 2.699 meters, and the 90% confidence interval for sixteen year olds ranges between 0.196 meters and 3.322 meters. The confidence interval increases as age increases because the standard deviation increases with age causing the limits to widen (Lucey, 2002). Assuming that the management sticks to the age limits of the twelve year olds, i.e. stick to the range of between 0.396 meters to 2.699 meters, 7.56% (Z = (0.286-0.396)/0.5215) of the five year olds would not meet the threshold. I.e. they would be below the minimum height of 0.396 meters while 16.14% (Z = (3.322-2.699)/0.95)of the sixteen year olds would exceed the maximum height of 2.699 meters. However, if the management sticks to the sixteen year olds i.e. stick to the range of between 0.196 meters and 3.322 meters, they would be able to cater to at least 90% of the children who visit the theme park. Given these findings, the management should not be too restrictive on the age.
References
Curwin, J. & Slater, R. (2008). Quantitative methods for business decisions. London: South-
Western Cengage Learning.
Lucey, T. (2002). Quantitative techniques. London: Thomson.
