How To Calculate – Average, Median, Standard Deviation, Variance, Hypothesis testing

Since the customers have been complaining that the amount of soda in the bottles do meet the advertised amount of sixteen ounces, it has forced us to carry a random test to determine the credibility of that claim. To help us do so, bottles were picked on a random basis by the employees across all shifts and measurements taken for all the bottles. Out of the 30 samples that were taken, the measures of dispersion were then calculated as sown below:

AVERAGE/ MEAN

(14.5 + 14.6 + 14.7 + 14.8 + 14.9 + 15.3 + 14.9 + 15.5 + 14.8 + 15.2 + 14.1 + 14.2 + 14 + 14.9 + 14.7 + 14.5 + 14.6 + 14.8 + 14.8 + 14.6 + 15 + 15.1 + 15 + 14.4 + 15.8 + 14 + 16 + 16.1 + 15.8 + 14.5) / 30

446.1/30

AVERAGE = 14.87

 

MEDIAN

The median value is the figure or value that creates a separation between the lower half and the higher half of a given set of numbers. In this case, the median value will be found by taking an arrangement of the value of the ounce from the smallest to the largest then finding the middle value that separates the two halves.  The median value is used to show the most resistant statistic of the values or data collected because it gives the 50% breakdown point. For the value of the pounces collected in the random sampling, the median value will be calculated as follows:

14, 14, 14.1, 14.2, 14.4, 14.5, 14.5, 14.5, 14.6, 14.6, 14.6, 14.7, 14.7, 14.8,   (14.8, 14.8,)   14.8, 14.9, 14.9, 14.9, 15, 15, 15.1, 15.2, 15.3, 15.5, 15.8, 15.8, 16, 16.1,

The two values placed in parentheses do fall in the middle. Since there are two values in the middle of the pack, the median will be calculated by summing them up and then dividing the answer by two

14.8 + 14.8 = 29.6

29.6/ 2 = 14.8

STANDARD DEVIATION

The value of the standard deviation is used to measure how the values that are given are spread out. For this case, we will measure the spread of the values of the bottle ounces. In order to get the value of the standard deviation, the value of the variance has to be first computed then later the value of standard deviation will be generated.

The variance is calculated by getting the average of the squared differences from the mean

 

(14-14.87)2 = 0.7569

(14 – 14.87)2 = 0.7569

(14.1- 14.87)2 = 0.5929

(14.2 – 14.87)2 = 0.4489

(14.4 – 14.87)2 = 0.2209

(14.5 -14.87)2 = 0.1369

(14.5 – 14.87)2 = 0.1369

(14.5 – 14.87)2 = 0.1369

(14.6 -14.87)2 = 0.0729

(14.6 – 14.87)2 = 0.0729

(14.6 – 14.87)2 = 0.0729

(14.7- 14.87)2 = 0.0289

(14.7 – 14.87)2 = 0.0289

(14.8 – 14.87)2 = 0.0049

(14.8 – 14.87)2 = 0.0049

(14.8 – 14.87)2 = 0.0049

(14.8 – 14.87)2 = 0.0049

(14.9 – 14.87)2 = 0.0009

(14.9 – 14.87)2 = 0.0009

(14.9 – 14.87)2 = 0.0009

(15 – 14.87)2 = 0.0169

(15 – 14.87)2 = 0.0169

(15.1 – 14.87)2 = 0.0529

(15.2 – 14.87)2 = 0.1089

(15.3 14.87)2 = 0.1849

(15.5 – 14.87)2 = 0.3969

(15.8 – 14.87)2 = 0.8649

(15.8 – 14.87)2 = 0.8649

(16 – 14.87)2 = 1.2769

(16.1 – 14.87)2 = 1.5129

8.783

Variance = 8.783/ 30 = 0.2927

Standard deviation of the values will be found by getting the square root of the variance

Standard deviation = 0.5410

Hypothesis testing

The null hypothesis is that 95% of the bottles contains less than sixteen ounce

A significance value of 0.85 will be chose for this work.

To determine the statistic probability, the value of the mean is divided by the original required value of sixteen ounce. The result gives 0.929375 which is close to the significance value. Since the value of 0.929375 is greater than the significance value, then the null hypothesis will be accepted.

Based on the calculations conducted above, it can clearly be seen that the majority of the bottles contained sodas of less than sixteen ounce. The primary cause may have been due to the high speed of the conveyor belt thereby not allowing enough time for the bottles to fill up. Secondly, it may be due to the low timings that have been set for the filling taps.

To avoid such deficits in the future, the speed of the conveyer belt will be adjusted so that it gives enough time for the bottles to fill up appropriately. Secondly, the timings of the filling taps will also be adjusted so that they release the sodas within the correct time.


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