Cholesterol Variable Science/Health students Assignment 2

Cholesterol

BloodSugar: Fasting blood sugar However, in this question, we will only need the Cholesterol variable. Suppose we wish to use a One-sample t-test to determine whether the true average cholesterol for the population associated with this sample of patients is different . What follows is a series of questions related to testing this hypothesis.

Carry out the necessary analysis using either jamovi (Sci/Health students) or R (Data Science students) and answer the following questions. Where relevant, round your answers to THREE decimal places.

1. What is the sample mean cholesterol, ?

2. What is the sample standard deviation, s?

3. What is the standard error, se?

Sci/Health Students: To visualise the data, create a histogram of Cholesterol with a density curve overlaid. Important note: Save your plot; in the next question you will have a chance to upload your plot. You do not have to answer any questions about your histogram until the next question.

Data Science Students: To visualise the data, create a histogram of Cholesterol with a normal curve overlaid. Important note: Save your R code used to create the plot, and save your plot. In the next question you will have a chance to upload your plot and you will also need to input your code. You do not have to answer any questions about your histogram until the next question. 

Question 1

• Mean = 243
• SD = 57.9
• SE = 9.93

Based on the Shapiro-Wilk test alone, we can assume normality since we have p = 0.149 In addition, since we have n = 34 the Central Limit Theorem can be applied. Therefore in conclusion, based on consideration of the Shapiro-Wilk test and Central Limit Theorem, we can assume normality.
df = 33
test statistic = 1.32
p-value = 0.195
CI = (-7.05, 33.3)

Since we have p = greater than, alpha = 0.05, we do not reject H₀ and therefore cannot conclude that the average cholesterol for this population is different from 230. Further supporting this conclusion is the fact that the confidence interval does contain 0.

Question 2

The histogram of the Cholesterol variable with an overlaid density curve provides some interpretations. It can be observed that the data is not perfectly normally distributed. The distribution is right skewed. At Cholesterol value of 200, there is a concentration of observations. The shape of the curve lacks symmetry and the curve does not show bell-curve characteristics which is typical of a normal distribution. However, the sample size is 34, and it is sufficiently large for applying the Central Limit Theorem. This means that even if the data does not appear to be normal, the sampling distribution of the sample mean will be approximately normal.

Question 3

Since we have p = greater than, α = 0.05, we do not reject H₀ and therefore cannot conclude that the difference in average cholesterol between the two groups is statistically significant. Further supporting this conclusion is the fact that the confidence interval does contain 0.

Question 4

Yes, there is a visual concern regarding the equal variance assumption when looking at the boxplot obtained. The following observations can be noted from the box plot. The “No” group has a much wider spread in cholesterol levels compared to the “Yes” group. The interquartile range as well as the length of the whiskers for the “No” group are both visibly larger. This indicates that the variance is higher in the “No” group than in the “Yes” group. The boxplot also indicates that the variances are unequal between the groups. This violates the equal variances assumption required for the standard independent samples t-test.

Question 5

0.105, weak (or very weak, if available)
A test for the significance of the correlation coefficient resulted in a p-value of 0.203 and a confidence interval of (-0.057, 0.260). Therefore, we cannot conclude that there is a statistically significant association between the two variables. This is because we have p = greater than, α = 0.05. Further supporting this conclusion is the fact that the confidence interval does contain 0.
0.011, very poor
0.275
Choose the best interpretation of the explanatory variable coefficient:
For each 1 mmHg increase in resting blood pressure (BP), the cholesterol level is estimated to increase by 0.275 mg/dL, on average.
244.288
(8). Since we have p = 0.203 > α = 0.05, we do not reject H₀. This means there is not enough evidence of a significant linear association between cholesterol and blood pressure.

Question 6

Linearity – The relevant plot for this assumption is the scatter plot of Cholesterol vs BP. It can be observed that the scatter plot shows a weak and dispersed linear trend. There is also a considerable spread around the fitted line. Thus, it can be said that linearity is likely not violated. While the relationship is weak (low R⊃2; = 0.0109), the plot does not show a clear curve or pattern that would indicate a non-linear relationship.

Constant Variance – The relevant plot for this assumption is the Residuals vs. Fitted Values plot. It can be observed that the residuals are scattered fairly evenly around zero, but the residuals also appear more spread out at higher fitted values. Thus, it can be said that there may be a mild violation of constant variance. The variability of residuals increases slightly with fitted values. This might indicate heteroscedasticity, but not severely.

Normality of Residuals – The relevant plot for this assumption is the Normal Q-Q Plot. It can be observed that most points lie close to the diagonal line, with some deviation in the tails especially at the extremes. Thus, it can be said that Normality assumption is almost met. A slight deviation in the tails usually appears in real data. But it can be concluded that the residuals appear approximately normally distributed.