2024 JPJC H1 Q11

A study on a large number of Type \(2\) diabetic patients is conducted. This study suggests that the probability that a randomly chosen Type \(2\) diabetic patient shows better control of blood glucose level after adopting a healthy daily diet is \(0.85\). A random sample of \(10\) Type \(2\) diabetic patients who adopted a healthy daily diet is taken and their blood glucose levels are noted.

1. Find the probability that the number of Type \(2\) diabetic patients who shows better control of blood glucose levels is
   1. \(7\) or \(8\),[1]
   2. more than \(6\),[2]
   3. at most \(8\) given that the number of Type \(2\) diabetic patients who shows better control of blood glucose levels is more than \(6\).[2]

It is known that the mean fasting blood glucose levels of Type \(2\) diabetic patients is \(7\) millimoles per liter (mmol/L). Researcher \(A\) has developed a particular diet which he claims will help in reducing blood glucose levels for Type \(2\) diabetic patients. A large number of Type \(2\) diabetic patients are given this diet. After a period of time of taking this diet, the fasting blood glucose levels, \(x\) mmol/L, of a random sample of \(50\) Type \(2\) diabetic patients are summarised as follows.

\[\sum(x-7) = -12.13 \quad \quad \sum(x-7)^2 = 111.28\]

2. Test, at \(10\%\) significance level, whether this information supports Researcher \(A\)'s claim.[6]

It is also known that the mean non-fasting glucose sugar levels is \(\mu_0\) mmol/L. Researcher \(B\) claims that this particular diet does not change the non-fasting blood glucose levels of Type \(2\) diabetic patients. After the diet has been given to the Type \(2\) diabetic patients for a period of time, it is found that the population standard deviation of the non-fasting blood glucose levels is \(2.3\) mmol/L. Another random sample of \(50\) Type \(2\) diabetic patients is taken and has mean non-fasting blood glucose levels of \(8.5\) mmol/L.

3. Find the set of possible values of \(\mu_0\) for Researcher \(B\)'s claim to be not supported at the \(5\%\) level of significance.[4]