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Skewness-drinkofwater.docx
Statistics
Juan David Solis
Hypothesis:
According to Heathline.com say that men older than 19 years must drink an average 13 cups of water per day. I believe that the actual mean is less than that.
Data collection:
The population is students from Johnson and Wales and the sample is the 50 Johnson and Wales students.
Ratio:
Ratio because it is highest level of measurement and has a true zero.
Skewness is a measure of asymmetry of a data set relative to the mean. A data set in which the mean is greater than the median is skewed to the right while a data set in which the mean is less than the median is said to be skewed to the left. We can also determine the skewness of a data set by observing its graph or by calculating the skewness coefficient.
For the data set of the number of glasses of water drank per day, we can determine the skewness by first generating a histogram using Excel. From the shape of the histogram generated by Excel, it seems that the data is slightly skewed to the right. We also compare the mean and the median, and notice that the mean is 8.2, which is greater than the median which is 8.0, and therefore indicating a slight skewness to the right. We can also determine the skewness by observing the skewness coefficient in the Descriptive Statistics table generated by Excel’s Data Analysis Tool Pak. The coefficient of skewness is 0.5444 (to 4 decimal places). We notice that the value of the skewness coefficient. We determine if this skewness coefficient is statistically significant for the distribution. Data is generally said to be highly skewed if it has a coefficient of skewness that is outside the range of -1 to +1 (Ekstrom, 2012; Rayner, 1995). Significantly right-skewed data have coefficient of skewness values more than +1 while significantly less-skewed data have coefficient of skewness less than -1. (Rayner, 1995). The coefficient of skewed data in this case is within the range for a data set that is not skewed.
The value of the coefficient of skewness in this case is not statistically significant. Since the coefficient is a positive value but less than +1, the conclusion we can make is that the data is not skewed and is approximately normal. If we became stricter with our conclusion, then we can say that the data is only slightly skewed to the right (slightly positively skewed).
Confidence Interval
A confidence interval gives a range of values with a specified probability that the value of a parameter lies within it with a certain probability. It has the lower and upper limits. For the case of the number of cups of water drunk by people over the age of 19 years, we determine the 95% confidence interval. The shape of the graph indicates this distribution is approximately normal. The sample size is also big (greater than 30), hence we can use a z-distribution as it approximates a t-distribution for this large sample size.
The formula for the confidence interval is:
z*
For the 95% confidence interval, the z-value is 1.96
In this case, = 8.2
s=2.595
n=50
Substituting:
The confidence interval is:
8.21.96*
= (7.48, 8.92)
There is a 95% probability that the mean of the number of glasses consumed by the students of Johnson and Wales who are over 19 years of age lies between 7.48 and 8.92 glasses.
Conclusion
The confidence interval may be used to determine if the population mean for the number of cups drank by those over 19 years is 13 glasses as recommended. The 95% confidence interval for the mean was found to be in the range (7.48, 8.92). We are 95% confident that the population mean will be in the range (7.48, 8.92) cups. We check if the recommended number of cups is in the interval or not. In this case, 13 cups is not within the interval but is even higher than the upper limit of the confidence interval. The conclusion is that the true mean water consumption by those over 19 years is less than the recommended 13 cups. We have sufficient evidence that the number of cups being drunk by those over 19 years old is less than 13 cups.
Other descriptive statistics measures such as the mean and the standard deviation also indicates that the water drank by the students of Johnson and Wales is likely to be less than 13 cups. For example, the mean of 8.2 is far less than 13. Most of the 50 values in the sample are far less than 13 cups. Therefore, from observing the sample data alone, we are sure that the sample clearly indicates that the population in this case does not meet the requirement of 13 cups per day.
The practical significance indicates that people over the age of 19 are not drinking the recommended 13 cups of water per day. Due to the fact that the confidence interval does not contain 13, we can be confident that there is less water consumption by those over 19 years of age. Therefore, campaign efforts should be made to encourage people to drink more water to meet the required threshold by health agencies. More studies need to be done to compare the results of this study with the others to arrive at a more precise conclusion. For example, studies from other populations such as in various cities should be conducted to verify the results of this study and establish if the results can be generalized to other populations.
References
Ekstrom, M. (2012). A general measure of skewness. Statistics & Probability Letters, 82(8),
1559-1568.
McCune, S.K. (2010). Statistics. New York, NY: McGraw-Hill
Rayner, J.C.W. (1995). Interpreting the skewness coefficient. Communications in Statistics –
Theory and Methods, 24(3), 593-600.
Conclusion
The hypothesis tests if the water consumption for those who are older than 19 years is the recommended 13 cups of water or less. The following is the hypothesis:
H0: µ=13
H1: µ<13
We follow the formal statistical procedures to test this.
The population is students from Johnson and Wales and the sample is the 50 Johnson and Wales students. The level of measurement is ratio because it has a true zero and is the highest level of measurement. We may need to determine the skewness of the distribution to find out the shape of the distribution. As determined before, the skewness coefficient and the shape of the histogram indicates that the distribution of the data is approximately normal. The skewness coefficient was not statistically significant, and therefore we could conclude that the distribution is not skewed. Therefore, we will use the normal distribution for this test. We also use the z-test since the sample size is 50 and is more than 30, we can assume that this distribution is approximately normally distributed (McCune, 2010). We can therefore, use the z-statistic as it will approximate the t-distribution for this sample size. We test at α=0.05. This is a left-tailed test
Z=
Substituting,
=-13.08 (to 2 dp)
The critical value is -1.645. It is clear that the test statistics is far much less than the critical value.
We have sufficient evidence that the number of cups being drunk by those over 19 years old is less than 13 cups.
The practical significance indicates that people over the age of 19 are not drinking the recommended 13 cups of water per day. Therefore, campaign efforts should be made to encourage people to drink more water to meet the required threshold by health agencies. More studies need to be done to compare the results of this study with the others to arrive at the a more precise conclusion.
