Statistics: What is the probability of type I error?

• sanctifier
In summary, the probability of type I error is approximately 0.0480 and the probability of type II error is approximately 0.8585.
sanctifier

Homework Statement

X is a random variable of binomial distribution of parameter n=10 and unknown parameter p. Hypotheses are given as follows:

$H_0 \;\; : \;\; p=0.6$

$H_1 \;\; : \;\; p \neq 0.6$

Suppose rejection region for $H_0$ is $\{X \leq 1\} \cup \{X \geq 9\}$

Question 1: What is the probability of type I error?

Question 2: If $H_1$ is changed to "$H_1 \;\; : \;\; p =0.3$", then what is the probability of type II error?

Homework Equations

Binomial Distribution of parameters n and p: $f(x) = \binom{n}x p^x(1-p)^{n-x}$

The Attempt at a Solution

$P(X \leq 1,\;\; X \geq 9|p=0.6)=1-P(2\leq X \leq 8|p=0.6)= 1-\sum_{k=2}^8 \binom{10}k 0.6^k(1-0.6)^{10-k} \approx 0.1689$

$P(2 \leq X \leq 8|p=0.3) = \sum_{k=2}^8 \binom{10}k 0.3^k(1-0.3)^{10-k} \approx 0.617$

Last edited:
Help!

Does anyone know the correct solution?

sanctifier said:
Help!

Does anyone know the correct solution?

I get 0.04803512320 ≈ 0.0480 for question 1 and .8505479682 ≈ 0.8585 for question 2.

1. What is a type I error in statistics?

A type I error, also known as a false positive, is a statistical error that occurs when a null hypothesis is rejected when it is actually true. This means that the test results falsely indicate that there is a significant effect or relationship when in reality there is none.

2. How is the probability of type I error calculated?

The probability of type I error is calculated by determining the significance level, denoted as α (alpha), which is the maximum acceptable probability of making a type I error. This is typically set at 0.05 or 0.01. The probability of type I error is then equal to the significance level.

3. What factors can affect the probability of type I error?

The probability of type I error can be affected by the sample size, significance level, and the variability of the data. A larger sample size and a lower significance level can decrease the probability of type I error, while a smaller sample size and a higher significance level can increase the probability of type I error.

4. How does type I error relate to type II error?

Type I error and type II error are two types of statistical errors that are inversely related. Type II error, also known as a false negative, occurs when a null hypothesis is not rejected when it is actually false. In other words, a type II error is the failure to detect a significant effect or relationship when it actually exists. Therefore, as the probability of type I error decreases, the probability of type II error increases and vice versa.

5. How can type I error be minimized?

The best way to minimize the probability of type I error is to carefully choose the significance level and conduct a power analysis to determine an appropriate sample size. It is also important to carefully design the study and select appropriate statistical tests to avoid making incorrect conclusions based on the results.

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