Answer :
Final answer:
The student's question involves the normal distribution of heat evolved in calories per gram for a cement mixture, which is a topic in Physics, particularly within the domain of calorimetry. Calorimetry is used to measure energy changes in substances, interchanging units like calories and joules, and requires understanding of the specific heat capacity.
Explanation:
Understanding Calorimetry in Physics
The student's question pertains to the heat evolved in calories per gram of a cement mixture, which follows a normal distribution with a mean of 99.8 and a standard deviation of 5.2. This problem is rooted in the principles of thermodynamics and calorimetry, which are branches of Physics. Calorimetry is the science of measuring the heat of chemical reactions or physical changes. The unit 'calorie' is used to express energy, and it is defined as the amount of heat required to raise the temperature of one gram of water by one degree Celsius.
The conversion between different energy units, such as calories and joules, is a common task in calorimetry. For example, the specific heat capacity of a substance, which is measured in joules per kilogram per Celsius degree (J/kg/°C), is a pivotal factor in calculating energy changes during temperature variations. Water, for instance, has a specific heat capacity of approximately 4,184 J/kg/°C, closely associated with the kilocalorie definition. In calorimetry experiments, such as those involving food like peanuts, the number of kilocalories per gram is determined by burning the sample and measuring the heat transfer, often to water.
- (A) If the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α is 0.09296.
- (B) For the case in which the true mean heat evolved is 103, β is 0.04648.
What is heat?
- The transfer of kinetic energy from one medium or object to another, or from an energy source to a medium or object, is referred to as heat.
- Energy can be transferred in three ways: radiation, conduction, and convection.
- When kings weren't hiding in oak trees or being beheaded in the 1600s, they enjoyed a game of horse racing.
- And when they took their horse for a gallop to prepare it for a race, they called it heat because they were obviously warming the horse up.
To find the answers to the given situation:
Use n = 5, everything else held constant:
(A) To find the type I error probability α:
- P( X ≤ 98.5) + P( X >101.5)
- P(X-100/(2/√5) ≤ 98.5-100/(2/√5)) + P(X-100/(2/√5)>101.5-100/(2/√5))
- P(Z ≤ −1.68) + P(Z > 1.68)
- P(Z ≤ −1.68) + (1 − P(Z ≤ 1.68))
- = 0.04648 + (1 − 0.95352) = 0.09296
So, if the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α has is 0.09296.
(B) To find β:
- β = P(98.5 ≤ X ≤ 101.5 when µ = 103)
- P(98.5-103/(2/√5) ≤ X-103/(2/√5) ≤ 101.5-103/(2/√5))
- P(−5.03 ≤ Z ≤ −1.68) = P(Z ≤ −1.68) − P(Z ≤ −5.03)
- = 0.04648 − 0 = 0.04648
So, for the case in which the true mean heat evolved is 103, β is 0.04648.
Therefore, answers to both the questions are shown.
Know more about the heat here:
https://brainly.com/question/19666326
#SPJ4
The correct question is given below:
The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100, and the standard deviation is 2. You wish to test H0 : µ = 100 versus H1 : µ 6= 100 with a sample size of n = 5 specimens.
(a) If the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α.
(b) Find β for the case in which the true mean heat evolved is 103.