Statistics for Data Science II End Term: 7 August 2022 (May 2022 term)

The standard deviation of the length of bolts manufactured by a machine was 1 mm. A new machine with a supposedly lesser standard deviation is installed. A random sample of 100 bolts from the new machine showed a standard deviation of 0.5 mm. Based on this data, what is your conclusion at a significance level of 0.05, on the standard deviation of length of bolts manufactured by the new machine? Choose the correct options from the following (Select all that apply):

The average life of a battery is 4 years, with a standard deviation of 1 year. Assuming that the lives of the batteries follow approximately a normal distribution, find the probability that the mean life of a random sample of 30 such batteries falls between 3.8 and 4.2 years. Enter your answer correct to two decimal places.

Based on the above data, answer the given subquestions.

Choose the correct option(s) from the following (TX refers to the values taken by a random variable X):

Based on the above data, answer the given subquestions.

Calculate P(X1 = 0, X2 = 1 | X3 = 1).
Note: Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

Find the marginal distribution of Y .

Based on the above data, answer the given subquestions.

Compute E[Y ].
Note: Enter the answer correct to two decimal places.

Based on the above data, answer the given subquestions.

Compute E[Y | X = 0.5].
Note: Enter the answer correct to two decimal places.

Suppose the number of deceased people in a week because of a deadly disease in a country has the Poisson(λ) distribution. A researcher wants to estimate λ. From studies in other countries, prior mean should be 3, and the prior standard deviation 1. He decides to use the Gamma(α, β) prior that matches his prior mean and standard deviation. The number of deceased people recorded over the next 10 weeks are: 3, 5, 7, 0, 1, 7, 4, 9, 6, 4.
Based on the above data, answer the given subquestions.

Find the value of α and β.

Suppose the number of deceased people in a week because of a deadly disease in a country has the Poisson(λ) distribution. A researcher wants to estimate λ. From studies in other countries, prior mean should be 3, and the prior standard deviation 1. He decides to use the Gamma(α, β) prior that matches his prior mean and standard deviation. The number of deceased people recorded over the next 10 weeks are: 3, 5, 7, 0, 1, 7, 4, 9, 6, 4.
Based on the above data, answer the given subquestions.

Find the posterior mean of λ for the given sample. Enter the answer correct to two decimal places.

The time (in minutes) required to repair electronic gadgets by an electrician is an exponential random variable with unknown parameter λ. Consider a sample (in minutes)
10, 11, 25, 1, 4, 5, 8, 35, 14, 15 from his previous repairing time.
Based on the above data, answer the given subquestions.

Find the method of moments estimate of λ for the given sample. (Enter the answer correct to three decimal places.)

The time (in minutes) required to repair electronic gadgets by an electrician is an exponential random variable with unknown parameter λ. Consider a sample (in minutes)
10, 11, 25, 1, 4, 5, 8, 35, 14, 15 from his previous repairing time.
Based on the above data, answer the given subquestions.

Find the maximum likelihood estimate of λ for the given sample. (Enter the answer correct to three decimal places.)

The time (in minutes) required to repair electronic gadgets by an electrician is an exponential random variable with unknown parameter λ. Consider a sample (in minutes)
10, 11, 25, 1, 4, 5, 8, 35, 14, 15 from his previous repairing time.
Based on the above data, answer the given subquestions.

Using a Uniform[0,1] prior, find the posterior mean of the given sample. (Enter the answer correct to three decimal places)

Suppose a principal states that the students studying in her school have an average IQ score equal to 98.5. A reporter suspects that the average may be lower, possibly 98 and took a sample of 100 random students from the school and observed the average IQ score to be 98.29. Assume the IQ scores of the students are normally distributed with standard deviation 1.2.
Based on the above data, answer the given subquestions.

Define null hypothesis and alternative hypothesis.

Suppose a principal states that the students studying in her school have an average IQ score equal to 98.5. A reporter suspects that the average may be lower, possibly 98 and took a sample of 100 random students from the school and observed the average IQ score to be 98.29. Assume the IQ scores of the students are normally distributed with standard deviation 1.2.
Based on the above data, answer the given subquestions.

Find the P -value. Enter the answer correct to two decimal places.

Suppose a principal states that the students studying in her school have an average IQ score equal to 98.5. A reporter suspects that the average may be lower, possibly 98 and took a sample of 100 random students from the school and observed the average IQ score to be 98.29. Assume the IQ scores of the students are normally distributed with standard deviation 1.2.
Based on the above data, answer the given subquestions.

Choose the set of correct options.

Answer the given subquestions.

Answer the given subquestions.

Select the steps from the following options, that you will use for finding the P- value in the previous question.
Note: This question is optional. We will check your answer to this question if you make a mistake in the previous one.