In: Statistics and Probability
is the graph showing normal distribution. explain
Tme of the day |
Aggregation values |
|||||||||||||||
Day-time (n = 16) |
6.23 |
6.91 |
6.35 |
6.29 |
6.45 |
6.30 |
6.60 |
6.54 |
6.64 |
6.90 |
6.11 |
7.28 |
6.93 |
7.89 |
7.21 |
6.90 |
Night-time (n = 16) |
6.41 |
5.98 |
6.25 |
6.03 |
6.57 |
6.25 |
6.51 |
6.50 |
6.50 |
6.41 |
6.70 |
6.03 |
6.60 |
6.77 |
6.88 |
6.93 |
Data analysis.
is the above graph shows normal distribution or not. explain n detail
Sampling technique: The data is probably an average for multiple readings (say 4 or 5 readings) for each of the 16 patients during day time and during night time. Taking multiple readings would reduce errors and get a closer to true value for each patient.
Below is the data summary:
Time of the day |
Average | Std Dev | Median | Minimum | Maximum |
Day-time (n = 16) | 6.720625 | 0.469332 | 6.62 | 6.11 | 7.89 |
Night-time (n = 16) | 6.4575 | 0.292313 | 6.5 | 5.98 | 6.93 |
Both mean and median are greater for Day-time and less for Night-time data. The Minimum and Maximum are also greater. In 12 out of 16 patients, that is 75% cases, the Day-time frequency is greater than Night-time frequency.
Below is the histogram and QQ-plots vs. normal distribution. The distribution are close but not normal. The first is skewed to right and second shows slight left skew with a much higher mode.
Comparison of Variances:
Standard Error for Day-Time = 0.469332
Standard Error for Night-time = 0.292313
Standard Error is an estimate for square root of Variance.
Therefore Day-time has greater variance. Test of Hypothesis for equality of two variances is the F-test.
The Test Statistic is
F = Var(1) / Var(2)
where Var(1) and Var(2) are the sample variances in groups 1 and 2.
The degrees of freedom for F are N_1 - 1 and N_2-1 where N_1 and N_2 are sample sizes of groups 1 and 2.
Null Hypothesis is that Day-time variance = Night-time variance
Alternative Hypothesis for 1-tailed is that Day-time variance is greater.
F statistic for the given data is
F = (0.469332*0.469332) / (0.292313*0.292313)
F (df1 = 15, df2 = 15) = 2.577899
With alpha = 0.05 for a 1-tailed distribution
check if F is greater than the critical value.
F_critical_upper_limit = 2.40
Reject null hypothesis of equality of variances if F > 2.40
F-value = 2.577899
We can reject the null hypothesis at significance level alpha = 0.05 and accept alternative hypothesis that the Day-Time observations show greater variance. This is also clear from the histograms, because the Day-time data shows greater spread (range for day-time = 1.78 vs. range for night-time = 0.95), whereas Night-time shows a sharp mode and is centered around the mode.