Q1. What is the probability that the sample mean support for corporate sustainability (x̄) exceeds 65, given a population mean of 68, standard deviation of 27, and a sample size of 45?

Background

Topic: Sampling Distributions and Probability

This question tests your understanding of the sampling distribution of the sample mean and how to calculate probabilities using the normal distribution.

Key Terms and Formulas

• Population mean (): The average value in the population.

• Population standard deviation (): The spread of values in the population.

• Sample mean (): The average value in your sample.

• Sample size (): Number of observations in the sample.

• Standard error ():

• Z-score:

Step-by-Step Guidance

1. Identify the known values: , , , and you are interested in .

2. Calculate the standard error (SE) using .

3. Compute the Z-score for using .

4. Use the standard normal table to find the probability that is greater than your calculated value.

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Q2. In a sample of 500 adults, what is the probability that the sample proportion of cord-cutters () is less than 0.12, given that the population proportion is 0.15?

Background

Topic: Sampling Distribution of the Sample Proportion

This question tests your ability to use the normal approximation for the sampling distribution of a sample proportion and calculate probabilities.

Key Terms and Formulas

• Population proportion (): The true proportion in the population.

• Sample proportion (): The proportion in your sample.

• Sample size (): Number of observations in the sample.

• Standard error for proportions:

• Z-score:

Step-by-Step Guidance

1. Identify the known values: , , and you are interested in .

2. Calculate the standard error using .

3. Compute the Z-score for using .

4. Use the standard normal table to find the probability that is less than your calculated value.

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Q3. Construct a 95% confidence interval for the true mean resale value of a 5-year-old foreign sedan, given a sample of 17 cars with a mean of $13,800 and a standard deviation of $600.

Background

Topic: Confidence Intervals for the Mean (Small Sample, Unknown Population Standard Deviation)

This question tests your ability to construct a confidence interval for the mean using the t-distribution.

Key Terms and Formulas

• Sample mean (): The average resale value in your sample.

• Sample standard deviation (): The spread of resale values in your sample.

• Sample size (): Number of cars in the sample.

• Degrees of freedom ():

• Standard error:

• Confidence interval:

Step-by-Step Guidance

1. Identify the known values: , , .

2. Calculate the standard error: .

3. Determine the degrees of freedom: .

4. Find the critical t-value () for a 95% confidence interval with 16 degrees of freedom (use a t-table).

5. Set up the confidence interval formula: .

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Q4. Find a 90% confidence interval for the true proportion of crimes involving a firearm, given 380 out of 600 crimes involved a firearm.

Background

Topic: Confidence Intervals for a Proportion

This question tests your ability to construct a confidence interval for a population proportion using the normal approximation.

Key Terms and Formulas

• Sample proportion ():

• Sample size (): 600

• Standard error:

• Critical z-value for 90% confidence: (find from z-table)

• Confidence interval:

Step-by-Step Guidance

1. Calculate the sample proportion: .

2. Compute the standard error: .

3. Find the critical z-value () for a 90% confidence interval.

4. Set up the confidence interval formula: .

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Q5. (a) State the null and alternative hypotheses for testing whether the average time to fill out pension form ABC-5500 has been reduced from 63 hours. (b) Draw the standard normal and find the rejection region for the test. (c) Calculate the test statistic and conduct the test, given a sample of 72 with mean 62.7 hours and standard deviation 20 hours.

Background

Topic: Hypothesis Testing for the Mean (Large Sample)

This question tests your ability to set up hypotheses, determine rejection regions, and calculate test statistics for a one-sample z-test.

Key Terms and Formulas

• Null hypothesis (): The claim to be tested (no reduction in time).

• Alternative hypothesis (): The claim that time has been reduced.

• Test statistic: , where

• Rejection region: Based on significance level (), find the critical value from the z-table.

Step-by-Step Guidance

1. State the null and alternative hypotheses: , (since you are testing for a reduction).

2. Draw the standard normal curve and shade the rejection region in the lower tail (since it's a left-tailed test).

3. Find the critical z-value for your chosen (commonly 0.05 unless otherwise specified).

4. Calculate the standard error: .

5. Compute the test statistic: .

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Q6. Test the null hypothesis that the population mean is 4 against the alternative that it is not 4, given a sample of 8 with mean 5.2 and standard deviation 1.1. Use .

Background

Topic: Hypothesis Testing for the Mean (Small Sample, t-test)

This question tests your ability to perform a two-tailed t-test for the mean when the population standard deviation is unknown and the sample size is small.

Key Terms and Formulas

• Null hypothesis ():

• Alternative hypothesis ():

• Test statistic: , where

• Degrees of freedom:

• Critical t-value: Find from t-table for and

Step-by-Step Guidance

1. State the null and alternative hypotheses: , .

2. Calculate the standard error:

   .

3. Compute the test statistic: .

4. Determine the degrees of freedom: .

5. Find the critical t-value for (two-tailed) and .

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Q7. (a) State the null and alternative hypotheses for testing whether a new cancer screening method is more accurate than the current method (which fails 15% of the time). (b) Draw the standard normal and find the rejection region for . (c) Calculate the test statistic and conduct the test, given 8 failures out of 70 tests.

Background

Topic: Hypothesis Testing for a Proportion

This question tests your ability to perform a one-sample z-test for a proportion.

Key Terms and Formulas

• Null hypothesis ():

• Alternative hypothesis (): (if testing for improvement, i.e., fewer failures)

• Sample proportion:

• Standard error:

• Test statistic:

• Critical z-value for (one-tailed)

Step-by-Step Guidance

1. State the null and alternative hypotheses: , .

2. Draw the standard normal curve and shade the rejection region in the lower tail.

3. Find the critical z-value for (one-tailed).

4. Calculate the sample proportion: .

5. Compute the standard error: .

6. Calculate the test statistic: .

Try solving on your own before revealing the answer!