Answer:
1. To determine the population infection rate that would yield a 90% confidence in an accurate result for a test with 5% false positives, we can use the concept of conditional probability and Bayes' Theorem. The infection rate is the proportion of the population that is actually infected. We want to find the infection rate that corresponds to a 90% confidence level, given a test with a 5% false positive rate.
Let's define the following terms:
I: The event that a person is infected.
+T: The event that a person tests positive.
F+: The event that a person has a false positive result.
We know the following probabilities:
P(+T|I): The probability that a person tests positive given that they are infected is the sensitivity of the test. This is the "true positive rate."
Sensitivity = 1 (100%) since you want the test to accurately detect all infected individuals.
P(F+|¬I): The probability that a person has a false positive result given that they are not infected. This is the false positive rate.
False Positive Rate = 5% = 0.05.
We want to find the infection rate, P(I), that corresponds to a 90% confidence level:
P(+T|I) = 90% = 0.90 (since a 90% confidence level implies a 10% chance of an incorrect result).
Now, we can use Bayes' Theorem to find the infection rate:
P(I|+T) = [P(+T|I) * P(I)] / P(+T)
P(+T) can be calculated using the law of total probability:
P(+T) = P(+T|I) * P(I) + P(+T|¬I) * P(¬I)
P(+T|¬I) is the probability that a person tests positive given that they are not infected. It is equal to 1 minus the specificity of the test (the true negative rate). If the test has a 5% false positive rate, the specificity is 95%:
P(+T|¬I) = 1 - Specificity = 1 - 0.95 = 0.05.
P(¬I) is the probability that a person is not infected. Therefore, P(¬I) = 1 - P(I).
Now, we can calculate P(+T):
P(+T) = P(+T|I) * P(I) + P(+T|¬I) * P(¬I)
0.90 * P(I) = [1 * P(I)] / [0.90 * P(I) + 0.05 * (1 - P(I))]
Now, you can solve this equation for P(I). After solving, you will find the infection rate that corresponds to a 90% confidence level with a test having a 5% false positive rate.
2. Cryptographic hashes are a specific type of hash function designed to meet certain security requirements and are commonly used in cryptography and blockchain technology. To be considered a secure cryptographic hash, several features are required:
Deterministic: For the same input, a cryptographic hash function must always produce the same hash value.
Fast Computation: The hash function should be computationally efficient to process data quickly.
Pre-image Resistance: Given the hash value (output), it should be computationally infeasible to determine the original input.
Second Pre-image Resistance: Given an input, it should be computationally infeasible to find another input that hashes to the same value.
Collision Resistance: It should be computationally infeasible to find two different inputs that produce the same hash value.
Avalanche Effect: A small change in the input should result in a significantly different hash value.
Pseudorandomness: The hash values should appear random and not reveal any information about the original data.
Fixed Output Size: The hash function should produce a fixed-size output, regardless of the size of the input data.
Cryptographic hashes are a fundamental component of blockchain technology, contributing to data integrity and security. Here's how they maintain data integrity in a blockchain:
Data Integrity: Each block in a blockchain contains a cryptographic hash of the previous block (the "parent" block). This hash value acts as a digital fingerprint for the previous block's data. If any data in the previous block is altered, the hash of that block changes, which would be immediately detected.
Immutability: Once a block is added to a blockchain, it becomes extremely difficult to alter or remove any data within it. This ensures the historical integrity of the ledger, making it suitable for various applications, including cryptocurrencies and secure record-keeping.
Data Verification: Anyone with access to the blockchain can independently verify the data integrity by recomputing the hash values. This is a crucial aspect of blockchain transparency and trustworthiness.
In summary, cryptographic hashes play a pivotal role in maintaining data integrity in blockchains by creating an interconnected, tamper-evident chain of blocks. Any unauthorized changes to the data in a block would require altering all subsequent blocks, which is practically impossible due to the properties of secure cryptographic hash functions.
Step-by-step explanation: