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The exponential distribution is a type of probability distribution that is often used to model the time between independent events that happen at a constant rate. In this context, it is used to describe the response time of a computer system. The key parameter of an exponential distribution is \( \lambda \), which is the rate parameter. In this exercise, the rate parameter \( \lambda \) is 1, simplifying the calculations, as we are given the probability density function (pdf) of \( X \) under the hypothesis \( H_0 \) as \( f_0(x) = e^{-x} \) for \( x \geq 0 \).

• This distribution has a mean of \( 1/\lambda \) and a variance of \( 1/\lambda^2 \).
• It is memoryless, meaning the probability of an event occurring in the future is independent of the past events.

The cumulative distribution function (CDF), \( F(x) = 1 - e^{-\lambda x} \), helps in determining the probability that a random variable \( X \) is less than or equal to \( x \). Here, it simplifies to \( F(x) = 1 - e^{-x} \) due to \( \lambda = 1 \). This feature is crucial in determining class intervals with equal probability in this chi-squared test scenario.

In hypothesis testing, class intervals are ranges into which observations are sorted for the purpose of comparing observed and expected frequencies. For the chi-squared test, these intervals must have equal probability under the null hypothesis.

In the context of this exercise with an exponential distribution, the cumulative distribution function (CDF) \( F(x) = 1 - e^{-x} \) is used to determine these intervals. Since we want five intervals, and each should cover 20% probability (0.2) under \( H_0 \), we solve \( F(x) = q \) for \( q = 0.2, 0.4, 0.6, 0.8 \):

• For \( q = 0.2 \), solve \( 1 - e^{-x} = 0.2 \) which gives \( x \approx 0.223 \).
• For \( q = 0.4 \), solve \( 1 - e^{-x} = 0.4 \) which gives \( x \approx 0.511 \).
• For \( q = 0.6 \), solve \( 1 - e^{-x} = 0.6 \) which results in \( x \approx 0.916 \).
• For \( q = 0.8 \), solve \( 1 - e^{-x} = 0.8 \) leading to \( x \approx 1.609 \).

Thus, the class intervals in this exercise are: - \([0, 0.223)\)- \([0.223, 0.511)\)- \([0.511, 0.916)\)- \([0.916, 1.609)\)- \([1.609, \infty)\)

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In this instance, we are testing the hypothesis that the response times of a computer system follow an exponential distribution. The null hypothesis \( H_0 \) assumes that the distribution is exponential with \( \lambda = 1 \).Steps in Hypothesis Testing:

• Formulate the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \).
• Select a significance level (\( \alpha \)), commonly 0.05, to determine the threshold for rejecting \( H_0 \).
• Calculate a test statistic from the sample data.
• Compare the test statistic to a critical value to decide whether to reject \( H_0 \).

The chi-squared test is used here to compare observed frequencies, obtained by sorting sample data into determined class intervals, against expected frequencies. A chi-squared statistic \( \chi^2 \) is computed and compared with a critical value to decide the fate of \( H_0 \). If \( \chi^2 \) is less than the critical value, we fail to reject \( H_0 \). In this exercise, \( \chi^2 = 3.25 \), which is less than the critical value 9.488 for 4 degrees of freedom at a 0.05 significance level, thereby failing to reject the null hypothesis.

Degrees of freedom (df) in statistical tests refer to the number of values or quantities that can vary in the final calculation of a statistic. For a chi-squared test, the degrees of freedom are calculated as the number of categories minus 1.

In this exercise, where the data is divided into five class intervals, the degrees of freedom are calculated as 5 (the number of categories) minus 1, which results in 4. Degrees of freedom are vital because they determine the critical value from the \( \chi^2 \) distribution tables. They reflect the number of independent comparisons that can be made between the observed and expected frequencies.

• The more class intervals, or more complex the model, the higher the degrees of freedom.
• Each degree of freedom represents an independent direction in which data can vary.

In our chi-squared test, with 4 degrees of freedom, we use this to find the critical value (9.488 with \( \alpha = 0.05 \)) against which the test statistic (\( \chi^2 = 3.25 \)) is compared. The degrees of freedom ensure that the test correctly assesses the hypothesis given the available data.