Applied Statistics and Probability for Engineers - Assignment Example

Step 1: The assumptions of normality of the populations from which the samples are derived are assumed to hold. Moreover, the samples are independent and standard deviations are known.
Step 2: We wish to test the hypotheses that;
H0: There is no statistically significant difference in mean between students majoring in Liberal Arts and other students on the average number of books read per year.
H1: There is a statistically significant difference.
Alternatively, this can be stated as;
H0: μ1 – μ2 = 0, where μ1=mean for Liberal Arts and μ2=mean for others.
H1: μ1 ≠ μ2
Step 3: We set α = 0.05
Step 4: Degrees of freedom are obtained as follows;
df1 (for Liberal Arts) = n1 – 1 = 236 – 1 = 235
df2 (for Others) = n2 – 1 = 321 – 1 = 320
dftotal = (n1 + n2 – 2) or (df1 + df2) = (235 + 320) = 555
Step 5: Since this is a 2-sided test, the critical region is given by t.025, 555= critical = 1.96
Step 6: Therefore, we compute the test statistic, t, as follows;

(Montgomery & Runger 2002, p. 339)
Where;
(xbar1 – xbar2) = Observed difference between sample means.
(μ1 – μ2) = Expected difference between population means if H0 is true.
= Estimate of the standard error of the difference between two sample means.
Where the pooled estimate of variance is given as;

Step 7: Therefore, t = 4.17 > critical = 1.96
Step 8: Decision: The null hypothesis (H0) is rejected.

Question 2
The results (as in table 1) indicate significant differences between the participant's age (M = 24.51, SD = 2.09) and the population mean of 25; t(186) = -3.18, p = .00. This implies that most of the participants of the present study had a lower age as compared to the average of the population age.
Table 1
Results for mean differences between the sample mean age and population mean age
N Mean SD t df Sig. 95% CI
Participant’s Age 187 24.51 2.09 -3.18 186 .00* [-.79, -.18]
Note. CI = confidence interval
*p < .05
Question 3
Methods
Participants
Before conducting the study, informed consent was first sought from the potential participants who mainly included members of the public. The study recruited a total of 187 participants of whom 76 were assigned the vignettes about a 13-year-old and 111 were assigned the vignettes about a 16-year-old.
Materials
The materials utilized in the present study included two vignettes, which described a crime. These vignettes were used to collect data from the participants.

The variable defendant’s age had two categories; 13-year-old defendants and 16-year-old defendants.
Results
Analysis
To compare the two groups, independent samples t-tests were conducted on the data obtained.
Findings
The results (as in table 2) indicated a lack of significant differences between participants who read a vignette about a 13-year-old (M = 1.89, SD = 1.21) and participants who read a vignette about a 16-year-old (M = 2.14, SD = 1.34); t(185) = 1.30, p = .19. This implies that the participants who read a vignette about a 13-year old were as likely to sentence the defendant to LWOP as those who read a vignette about a 16-year old.

Table 2
Results for independent samples T-Tests
N Mean SD t df Sig. 95% CI
Likelihood of sentencing to LWOP 111 2.14 1.34 1.30 185 .19 [-.13, .63]
Note. CI = confidence interval Read More