Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
A continuous stirred tank bioreactor operating at steady state has a working volume of 100 L and a feed flow rate of 20 L/h. The dilution rate (D) of this chemostat in h^-1 is:
Options:
(A) the probability of a type I error in a statistical test
(B) the error in estimating a sample standard deviation
(C) the standard deviation of a variable that follows standard normal distribution
(D) the standard deviation of distribution of sample means
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.26)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Biostatistics (Sampling Distributions & Standard Error)
**2. Detailed Step-by-Step Analysis**:
- In statistics, consider drawing repeated random samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$.
- According to the **Central Limit Theorem**, the sample means ($ar{X}$) form a sampling distribution with mean $\mu_{ar{X}} = \mu$.
- The standard deviation of this sampling distribution of sample means is defined as the **Standard Error of the Mean (SEM)**:
$ ext{SE}(ar{X}) = rac{\sigma}{\sqrt{n}}$
- Therefore, the standard error is formally defined as *"the standard deviation of distribution of sample means"*.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: The probability of a Type I error is the significance level $lpha$.
- **Option (B) [INCORRECT]**: This describes error in estimating sample variance, not the definition of standard error.
- **Option (C) [INCORRECT]**: The standard deviation of a standard normal distribution is identically equal to 1.
- **Option (D) [CORRECT]**: Standard error is precisely the standard deviation of the sampling distribution of sample means.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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