Descriptive Statistics and Probability - Basic Statistics

This document covers basic statistics concepts such as mean, median, mode, variance, standard deviation, correlation coefficient, chi-square test, confidence interval, t-test, z-score, and graphical analysis. It also includes tables and figures to support the analysis.

Running head: BASIC STATISTICS
Basic Statistics
Name of the Student:
Name of the University:
Author Note:

1DESCRIPTIVE STATISTICS AND PROBABILITY
Table of Contents
Solution 1:..................................................................................................................................2
Solution 2:..................................................................................................................................2
Solution 3:..................................................................................................................................3
Solution 4:..................................................................................................................................3
Solution 5:..................................................................................................................................4
Solution 6:..................................................................................................................................5
Solution 7:..................................................................................................................................6
Solution 8:..................................................................................................................................7
Solution 9:..................................................................................................................................7
Solution 10:................................................................................................................................9
Reference..................................................................................................................................10

2DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 1:
Table 1: Summary statistics of pre-treatment scores data
SCORE DATA
Mean 50.07
Median 41
Mode 37
Standard Deviation 22.90
Sample Variance 524.50
Range 79
Minimum 19
Maximum 98
Solution 1.a: The mean for the data is 50.07.
Solution 1.b: The median is 41.
Solution 1.c: The mode is 37.
Solution 1.d: The range is 79.
Solution 1.e: The variance is 524.50.
Solution 1.f: The standard deviation is 22.90.
Solution 2:
Table 2: Chi-square Test
P 0.00
Chi-square 51.4432
Chi-square at 0.05 15.51
Chi-square at 0.01 20.09
Solution 2.a: The Chi-square statistic is 51.4432.
Solution 2.b: At 0.05 level of significance, the null hypothesis is rejected as the critical
value of chi-square at 0.05 is less than the estimated chi-square value.

3DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 2.c: At 0.01 level of significance, the null hypothesis is rejected as the critical value
of chi-square at 0.01 is less than the estimated chi-square value (Pandis, 2016).
Solution 3:
Table 3: Correlation Coefficient
Correlation Coefficients
Pearson 0.83466
Solution 3.a: The Pearson r correlation coefficient for the data is 0.83466.
Solution 3.b: The correlation coefficient says that there is a strong correlation between the
two parameters as the coefficient value is close to 1.
Solution 4:
Solution 4.a:
Table 4: Pearson's coefficient (t test) at .05 significance level
Alpha 0.05
Tails 2
correlation 0.83
standard error 0.13
t 6.43
p-value 0.00
lower 0.56
upper 1.11
The p-value of the t-test is less than 0.05 that indicates that there is enough evidence
to reject the null hypothesis that there is no correlation between the two variables. This means
the alternative hypothesis is accepted that means there exist a positive correlation between the
two variables at 0.05 significance level (Koo & Li, 2016).

4DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 4.b:
Table 5: Pearson's coefficient (t test) at .01 significance level
Alpha 0.01
Tails 2
correlation 0.83
standard error 0.13
t 6.43
p-value 0.00
lower 0.46
upper 1.21
The p-value of the t-test is less than 0.05 that indicates that there is enough evidence
to reject the null hypothesis and accept the alternative hypothesis that means there exist a
positive correlation between the two variables at 0.01 significance level.
Solution 5:
Table 6: Margin of error and 95% confidence interval
Beck Depression- II Scores
Mean 39.60
Standard deviation 16.55
z-95% 1.96
n 20
Margin of error 7.25
Lower Confidence Interval 32.35
Upper Confidence Interval 46.85
Solution 5.a: The margin of error for beck depression- II score is 7.25.
Solution 5.b: The upper limits and lower limits of the confidence interval for beck
depression- II score at 95% significant level is 46.85 and 32.35.

5DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 5.c: At 95% significance level, it can be said that the average value of beck
depression-II score lies between 32.35 and 46.85.
Solution 6:
Solution 6.a:
Table 7: t-Test: Two-Sample Assuming Unequal Variances at 0.05 significance level
Pre-Treatment Post-Treatment
Mean 60 27.13
Variance 292.76 128.53
Observations 30 30
Hypothesized Mean Difference 0
df 50
t Stat 8.77
P(T<=t) one-tail 0.00
t Critical one-tail 1.68
P(T<=t) two-tail 0.00
t Critical two-tail 2.01
The t-stat is greater than the t-critical value at 0.05% significance level which indicate
that there is not enough evidence to accept the null hypothesis. Hence, the null hypothesis is
rejected and alternative hypothesis is accepted. Thus the conclusions is that the mean
difference between pre-treatment and post-treatment is statistically significant at 5%
significance level.
Solution 6.a:
Table 8: t-Test: Two-Sample Assuming Unequal Variances at 0.01 significance level
Pre-Treatment Post-Treatment
Mean 60 27.13
Variance 292.76 128.53
Observations 30 30
Hypothesized Mean Difference 0
df 50
t Stat 8.77

6DESCRIPTIVE STATISTICS AND PROBABILITY
P(T<=t) one-tail 0.00
t Critical one-tail 2.40
P(T<=t) two-tail 0.00
t Critical two-tail 2.68
The t-stat is greater than the t-critical value at 0.01% significance level. Hence, the
null hypothesis is rejected and alternative hypothesis is accepted. Thus the conclusions is that
the mean difference between pre-treatment and post-treatment is statistically significant at
5% significance level (Kim, 2015).
Solution 7:
Z-score for the charges of psychiatrist in South Dakota: 125−95
15 =2
Z-score for the charges of clinical psychologist in South Dakota: 120−95
15 =1.667
Z-score for the charges of clinical social worker in South Dakota: 100−95
15 =0.333
The lowest z-score offers the best deal. This means that the clinical social worker in
South Dakota offers the best deal (Di Gregorio et al. 2015).

7DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 8:
1 2 3
0
20
40
60
80
100
Hudson Test Score
Hudson Test Score
Session
Hudson Test Score
Figure 1: Single subject experimental design
Solution 9:
Figure 2: Visual analysis of level and SSED data stability
Solution 9.a: Condition X: The degree of improvement. This presents a positive change.
Solution 9.b: Condition X: The degree of stability. This presents no change.
Solution 9.a: Condition X: The degree of deterioration. This presents negative change.

8DESCRIPTIVE STATISTICS AND PROBABILITY
Figure 3: Graph to check the stability of given data
The data will be stable when the plotted data creates a straight line parallel to the x-
axis.
0 2 4 6 8 10 12
0
2
4
6
8
10
12
Scatter Diagram
Observations
Data Values
Mean 6.25
Standard Error 0.50
Median 6.95
Standard Deviation 2.24
Sample Variance 5.00
Range 7.4
Minimum 2.1
Maximum 9.5
Sum 125
Count 20

9DESCRIPTIVE STATISTICS AND PROBABILITY
Solution 10:
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0
0
1
2
3
4
5
6
7
8
9
10
Celerati on Line
Figure 4: Graphical analysis using Celeration line method
According to the celebration line method, the 50% points should lie above and below
the celeration line. However, there are 6 points above the line and 2 points below the line.
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0
0
1
2
3
4
5
6
7
8
9
10
11
12
2 standard deviati on method
Figure 5: Graphical analysis using 2-standard deviation band method
The intervention data points lies within the 2 standard deviation band.

10DESCRIPTIVE STATISTICS AND PROBABILITY
Reference
Di Gregorio, S., Del Rio, L., Rodriguez-Tolra, J., Bonel, E., Garcia, M., & Winzenrieth, R.
(2015). Comparison between different bone treatments on areal bone mineral density
(aBMD) and bone microarchitectural texture as assessed by the trabecular bone score
(TBS). Bone, 75, 138-143.
Kim, T. K. (2015). T test as a parametric statistic. Korean journal of anesthesiology, 68(6),
540.
Koo, T. K., & Li, M. Y. (2016). A guideline of selecting and reporting intraclass correlation
coefficients for reliability research. Journal of chiropractic medicine, 15(2), 155-163.
Pandis, N. (2016). The chi-square test. American journal of orthodontics and dentofacial
orthopedics, 150(5), 898-899.