Question

Can You please Answer the question using R studio and R cloud

Telomeres are complexes of DNA and protein that cap chromosomal ends. They consist of the same short DNA sequence TTAGGG repeated over and over again. They tend to shorted with cell divisions and with advancing cellular age, which will lead to chromosome instability or apoptosis (programmed cell death).

Eukaryotic cells have the ability to reverse telomere shortening by expressing telomerase, an enzyme that extends the telomeres of chromosomes. The level of telomerase activity is important in determining telomere length in aging cells and cancer cells.

One study of telomere length (measured as a ratio compared to a standard) were conducted and results were reported in the following table:

Individual	Cellular aging	Telomere length (relative ratio)	Telomerase activity (μmol/min)
1	1.8	1.75	0.68
2	2.2	1.60	0.54
3	3.0	1.65	0.51
4	4.7	1.33	0.42
5	5.1	1.35	0.45
6	6.0	1.28	0.41
7	7.5	1.20	0.44
8	8.2	1.19	0.46
9	8.9	1.31	0.50

1. Calculate the linear correlation between telomere length and the cellular aging. Is there asignificant relationship. (10 pts)
2. Use a different method from the one used in part (a) to calculate the linear correlation between telomere length and the cellular aging. Is there a significant relationship? (5 pts)
3. Estimate the slope and intercept of the least square regression line, with the telomerase activity as independent variable. (10 pts)
4. Using the t test to test the null hypothesis of zero slope at the significance level of a = 0.05. (10 pts)
5. What does MS _residual measure? What is the value of it in this study? (5 pts)
6. Calculate the R² statistic. What does it measure? (5 pts)
7. If a sample of cells has the telomerase activity 0.60 μmol/min, predict the length of telomere (as relative ratio) in those cells. (5 pts)

Answer 1

(a) r = -0.892

Source	SS	df	MS	F	p-value
Regression	43.1640	1	43.1640	27.28	.0012
Residual	11.0760	7	1.5823
Total	54.2400	8

The p-value is 0.0012.

Since the p-value (0.0012) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the relationship is significant.

(b) r = -0.892

Regression output				confidence interval
variables	coefficients	std. error	t (df=7)	p-value	95% lower	95% upper
Intercept	21.1488
X1	-11.2906	2.1617	-5.223	.0012	-16.4023	-6.1790

The p-value is 0.0012.

Since the p-value (0.0012) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the relationship is significant.

(c) y = 21.1488 - 11.2906*x

(d) The hypothesis being tested is:

H0: β1 = 0

H1: β1 ≠ 0

The p-value is 0.0012.

Since the p-value (0.0012) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the slope is significant.

(e) MS residual is a measure of how spread out these residuals are. The value is 1.5823.

(f) R2 = 0.796

R2 tells the explained variation in the data.

(g) y = 21.1488 - 11.2906*0.60 = 14.374