SuperFun Toys - The Best Toys for Kids of All Ages

SuperFun Toys is a company that deals with a variety of innovative children's toys. It has been observed that the pre-holiday period appears to be the best moment to introduce a new toy just before the December holiday. To do this, manufacturing should start in June or July in order to arrive at the store shelves by October. 

Normal distribution also known as the bell curve is commonly used in statistics to determine the interactions of mean and standard deviation of a give set of data. Normal distribution shows the likely outcome of a specific event. 

Question 1 

Normal distribution is defined and produced by two parameters that include the standard deviation (σ) and average or mean (µ). The density function of the distribution is ƒ(x)=1σ2π √ ҽ-1/2[(x-µ)/σ)]2 (Reid, 2017). The senior sales forecaster of SuperFun predicted 20,000 to be the projected demand for Weather Teddy. The company also estimated that there would be a 95% probability for the demand to range from 10,000 to 30,000 units. 20,000 is the mean and in order to find the normal distribution, standard deviation is important. SD is found by (30,000-10,000)/2 which equals 5,000. With the equation, the standard deviation is 5,102 units. The equation calculation is presented below: 

X = demand for Weather Teddy 

P(10,000<X<30,000) =

0.95 00 

P(10,000-20,000)/σ<(X-20,000)/σ < (30,000-20,000)/σ =

0.95 00 

Establishing the normal curve = (30,000-20,000)/σ =

1.96 00 

σ = (30,000-20,000)/1.96 00 = 10,000/1.96 00 =

5102

Normal distribution , in this case, is based on the mean and standard deviation and SuperFun senior sales forecaster should consider the standard deviation of 5,102 units before ordering the Weather Teddy for the upcoming holiday season.

Question 2 

The empirical rule dictates that the normal distribution should fall between the three standard deviations. For this rule to hold, three conditions must be met. First, 68% of the data must satisfy this rule. Secondly, about ninety-five percent (95%) of the data should be fall between two standard deviations, and thirdly 99.7% of it will be within the first three standard deviations of the distribution's average. In the given set of data above, a standard deviation of 5099.02 was seen and used to illustrate the difference between order quantities. 

Question 3 

The quantities are 15,000, 18,000 24,000 and 28,000. When these quantities of stock-out are calculated the probability of 15,000 is 34.13 % , 18,000 is 15.54%, 24,000 is 21.19 % , and 28,000 is 0.59 %. This show that of all the four quantities provided by the management 15,000 of stock-out has the largest probability of occurrence . Additionally , 28,000 of stock-out has the least chance of occurring . In this case, P (X > Y) = P (Z > (Y-20000)/5102). Z represents the standard normal distribution. 

Question 4 

The management has also asked for a revenue prediction for three various different cases, that is, pessimistic case , where sales is estimated to be 10,000 units, most probable scenario for 20,000 units; and optimistic case f or 30,000 units.

Uncertainty in decision making is experienced when the nature and timing of an event is unknown and the probability of the event occurring is also unknown. (Black, 2017). There are various different decision making methods. The pessimistic way is also known as maximum criterion. It is often assumed that the worst case scenario will happen and, therefore, measures are taken to reduce the likely loss. Pessimistic can also be described as worse-case-scenario. The optimistic method is also known as maxima criterion. The bullwhip effect is a phenomenon in the supply chain management that is used to analyse the inefficiencies that may exist within the channel. It is mostly applied when increasing production as one moves up the supply chain due to changing customer demands. It is commonly known that consumer demand changes from time to time thereby forming an amplitude when represented on a graph. It is this concept that the bullwhip effect was formulated. As one moves up the wave, it represents an increase in the quantity demanded, and therefore there is a need to change the supply pattern so as to meet the rising demand. 

The following shows the sales forecast. 

Computing the 70% 

P (X < Y) = 

0.70 

P (Z < (Y-20000)/ 5102) = 

0.700 

(Y-20000)/5102 = 

0.5244 

Y = 20000 + 5102 * 0.52440 = 20000 + 2675 = 

22675 

Probability that demand is greater than or equal to y= 0.70 

P (X ≥ y) =0.700 

P (X-μ/σ ≥ y- μ/σ) = 

0.700 

P (Z>y-20000/5102) = 

0.700 

P (X-μ/σ ≥ y-20000/5102) = 

0.700 

1-P (Z > y-20000/5102) = 

0.700 

P (Z≤ y-20000/5102) = 

0.300 

From normal distribution tables, we find that P (Z≤−0.5244) =0.30 

thus y-20000/5102= -0.5244 

Y=-0.5244×5102+20000 

Y=17324 

The quantity to be ordered is 17324 units 

Worst case in which sales equal 10000 units 

Expected sales price =10000 x24+240000 

Suggested order quantities is 17324 

Cost price is 17324 x16=277184 

Sale is only 10000 

So the stock out is 17324-10000=7324 units 

References 

Black, J. (2017). Business Statistics: For Contemporary Decision Making. John Wiley and Sons, Ince

Azzalini, A., & Valle, A. D. (1996). The multivariate skew-normal distribution. Biometrika , 83 (4), 715-726. 

Dixon, W. J., & Massey Frank, J. (2015). Introduction To Statistical Analsis . McGraw-Hill Book Company, Inc; New York.