Hypothesis Testing

Step 1: Understanding the Concept:
A Type I error occurs when we reject the null hypothesis ( ) when it is actually true. The probability of a Type I error is denoted by , also known as the significance level or the size of the test.

Step 2: Key Formula or Approach:
. The experiment follows a binomial distribution. Let X be the number of heads in 5 tosses. Then . The rejection rule is . We need to calculate under the assumption that is true, i.e., .

Step 3: Detailed Explanation:
Under , the number of heads X follows a binomial distribution with and .
The probability mass function (PMF) is .

The probability of a Type I error is the probability of rejecting , which is .
Calculate each probability:
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Sum the probabilities:
Step 4: Final Answer:
The probability of Type I error is .

Step 1: Understanding the Concept:
The power of a statistical test is the probability of correctly rejecting the null hypothesis ( ) when the alternative hypothesis ( ) is true. It is calculated as the probability of the observation falling into the critical region, assuming the parameter value from the alternative hypothesis.

Step 2: Key Formula or Approach:
Power = Given the critical region is , the power of the test is calculated using the distribution under .

Step 3: Detailed Explanation:
Under the alternative hypothesis, . The probability density function (PDF) of the population is: This is a uniform distribution on the interval [0, 2]. The critical region for rejecting is given as . The power of the test is the probability that the observation falls within this region, given that . We calculate this probability by integrating the PDF under over the critical region:
Step 4: Final Answer:
The power of the test is .

Step 1: Understanding the Concept:
A Type II error occurs when we fail to reject the null hypothesis ( ) when the alternative hypothesis ( ) is actually true. The probability of a Type II error is denoted by .

Step 2: Key Formula or Approach:
. The critical (rejection) region is given as . Therefore, the acceptance (fail to reject) region is its complement, which is . We need to calculate under the assumption that is true, i.e., .

Step 3: Detailed Explanation:
The population distribution is an exponential distribution with rate parameter . Under the alternative hypothesis, . The probability density function (PDF) is: The probability of a Type II error, , is the probability of the observation falling into the acceptance region , given that . We calculate this by integrating the PDF under over the acceptance region: This can be written as:
Step 4: Final Answer:
The size of Type II error is .

Step 1: Understanding the Concept:
The power of a test is the probability of rejecting when is true. The PDF is that of an exponential distribution with mean . We need to calculate the probability of the sample falling into the critical region , assuming the parameter value from .

Step 2: Key Formula or Approach:
1. Power = . 2. This is . 3. The sum of i.i.d. Exponential( ) random variables is a Gamma( ) variable. The given distribution has mean , so the rate is . 4. A Gamma distributed variable can be transformed into a Chi-squared variable using the relation: If , then .

Step 3: Detailed Explanation:
Under the alternative hypothesis , the random variables are i.i.d. from an exponential distribution with mean . The rate parameter is . Let . Since are i.i.d. Exp(1), their sum follows a Gamma distribution with shape parameter and rate parameter . So, . Now, we use the transformation to the Chi-squared distribution. If , then follows a Chi-squared distribution with degrees of freedom. The power of the test is calculated under . We transform this inequality to be in terms of the Chi-squared variable: Multiply both sides by 2: Since is a variable, the probability is:
Step 4: Final Answer:
The power of the test is .

Step 1: Understanding the Concept:
The size of a critical region (or test), denoted by , is the probability of rejecting the null hypothesis when is actually true. This is also known as the Type I error rate. The test is based on the maximum order statistic, .

Step 2: Key Formula or Approach:
1. The size of the test is . 2. In this case, . 3. We need the distribution of the maximum order statistic, , from a U(0, ) sample. The cumulative distribution function (CDF) of is , where is the CDF of a single U(0, ) variable.

Step 3: Detailed Explanation:
First, let's analyze the critical region under the null hypothesis . If , then all observations must be in the interval (0, 1). Consequently, the maximum observation, , must also be less than 1. This means the event is impossible under . Therefore, . The size of the test simplifies to: Now, we find the distribution of under . For a single observation from U(0,1), the CDF is for . For a sample of size , the CDF of the maximum order statistic is: The size is the probability that falls in the critical region, so we need to compute . This is given directly by the CDF of evaluated at .
Step 4: Final Answer:
The size of the critical region is .

Step 1: Understanding the Concept:
The problem requires a two-sample z-test for the difference between two population proportions. The null hypothesis is that the proportions are equal ( ). The test statistic measures how many standard errors the observed difference in sample proportions is from the hypothesized difference of zero.

Step 2: Key Formula or Approach:
The formula for the z-test statistic for two proportions is: where and are the sample proportions, and is the pooled proportion, calculated as:
Step 3: Detailed Explanation:
First, define the parameters for each city. City A (Sample 1): - , - City B (Sample 2): - , - Next, calculate the pooled proportion : Then . Now, calculate the z-statistic: Numerator: . Denominator term . Denominator: The calculated value is approximately 6.38. This is closest to option (B) 6.42. The minor difference may be due to rounding in the problem's expected answer.
Step 4: Final Answer:
The value of the test statistic is approximately 6.38, which is best represented by the option 6.42.

Concept:
In a Chi-square test for independence using a contingency table, the degrees of freedom are calculated using the formula: where:

• = number of rows
• = number of columns

Step 1: Identify the number of rows and columns.
Given contingency table: Thus,
Step 2: Apply the formula for degrees of freedom.

Step 3: State the conclusion.

Concept:
When comparing the means of two samples, the choice of test depends mainly on the sample size and knowledge of population variance.

• If the sample size is large ( ), the Z-test is generally used.
• If the sample size is small ( ) and the population variance is unknown, the Student's t-test is used.

The t-test is specifically designed for small samples and accounts for additional uncertainty in estimating the population variance.
Step 1: Identify the condition given in the question.
The problem states that the samples are small.
Step 2: Choose the appropriate statistical test.
For small sample means, the correct test is the Student's t-test.
Step 3: State the conclusion.

Concept:
In hypothesis testing, two types of errors may occur when making a decision about the null hypothesis.

• Type I Error: Rejecting the null hypothesis when it is actually true.
• Type II Error: Failing to reject the null hypothesis when it is actually false.

The probability of committing a Type I error is denoted by , which is also called the level of significance.
Step 1: Understand the situation given in the question.
The question states that the null hypothesis is rejected even though it is actually true.
Step 2: Identify the type of error.
This situation exactly matches the definition of a Type I Error.
Step 3: State the conclusion.