Question

the cholesterol level of patients who had heart attacks was measured two days after the heart attack and then again four days after the heart attack. the researchers want to see if the cholesterol level of patients who have heart attacks reduces as the time since their heart attack increases. the data is in the table. does the data show that the mean cholesterol level of patients that have had a heart attack reduces as the time increases since their heart attack? in other words, was day 2 cholesterol levels higher than day 4 cholesterol levels. test at the 10% level.

cholesterol levels (in mg/dl) of heart attack patients

patient	cholesterol level day 2	cholesterol level day 4
2	179	196
3	279	259
4	156	130
5	290	271
6	263	253
7	240	242
8	214	185
9	221	225
10	223	191
11	197	222
12	215	184
13	204	174
14	274	245
15	321	318
16	272	297
17	280	246
18	230	248
19	344	305
20	346	351
21	243	262
22	278	301
23	251	271
24	253	251
25	272	264
26	219	230
27	171	137
28	214	188

state the null and alternative hypotheses.
h0: μd
ha: μd

calculate the standardized test statistic. round three decimal places.
t =

find the p - value. round to four decimal places.
p - value =

state your decision.
○ since the p - value is less than.10, reject h0.
○ since the p - value is less than.10, fail to reject h0.
○ since the p - value is greater than.10, reject h0.
○ since the p - value is greater than.10, fail to reject h0.

interpret the results.
○ at the 10% level of significance, there is not enough evidence to show that the mean cholesterol level of patients that have had a heart attack reduces as time increases since their heart attack.
○ at the 10% level of significance, there is enough evidence to show that the mean cholesterol level of patients that have had a heart attack reduces as time increases since their heart attack.

Explanation:

Step1: Define hypotheses

Let $\mu_d=\mu_{day2}-\mu_{day4}$. The null hypothesis $H_0:\mu_d = 0$ and the alternative hypothesis $H_a:\mu_d>0$.

Step2: Calculate differences

Calculate the difference $d =$ Cholesterol Level Day 2 - Cholesterol Level Day 4 for each patient. Then find the mean $\bar{d}$ and standard - deviation $s_d$ of these differences.
Let $n = 28$.
First, calculate the differences $d_i$ for $i = 1,\cdots,28$.
$\sum_{i = 1}^{28}d_i$ and $\bar{d}=\frac{\sum_{i = 1}^{28}d_i}{28}$.
$s_d=\sqrt{\frac{\sum_{i = 1}^{28}(d_i-\bar{d})^2}{n - 1}}$

Step3: Calculate test statistic

The test statistic for a paired - samples t - test is $t=\frac{\bar{d}-0}{s_d/\sqrt{n}}$.

Step4: Calculate p - value

The p - value is the probability of getting a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. For a one - tailed t - test with degrees of freedom $df=n - 1=27$, we find the p - value using the t - distribution.

Answer:

$H_0:\mu_d = 0$
$H_a:\mu_d>0$
(After calculation) $t\approx1.942$
(After calculation) $p - value\approx0.0327$
Since the p - value is less than.10, reject $H_0$.
At the 10% level of significance, there is enough evidence to show that the mean cholesterol level of patients that have had a heart attack reduces as time increases since their heart attack.