Provide the logical reasoning for all questions.

1- Due to the capacity of laptops, a market analyst believes that the number of laptops per household has increased. In the past, themean number of laptops per household was 1.8 with a standard deviation of 0.3. To test his belief, the analyst takes a random sample of 37 households and finds a mean of 2.9 laptops. Let the significance level by 1%. Assume the conditions are satisfied. What is the value of the calculated test statistic? Please give your answer correctly rounded to two decimal places.

2- A company manufacturing computer chips wants to estimate the proportion of defective chips manufactured per day.A random sample of 1000 chips revealed 50 defects. Assume the conditions are satisfied. SHOW YOUR WORKINGS to four decimal places.

a) Create a 90% confidence interval.

b) Interpret your confidence interval.

c) Without calculating another confidence interval, and holding all else constant, would a 99% confidence interval be narrower, the same width or wider than your 90% confidence interval from part a)? Briefly explain.

3- The makers of an exercise device advertise that using their machine a few minutes per day results in an average weight loss of 5 kilograms during the first month. A consumer group believes the average weight loss would be less than this. The consumer group recruited 25 volunteers to use the product in accordance with the manufacturer's recommendations. They found an average weight loss of 4.8 kilograms with a standard deviation of 1.5 kilograms. You are told that the calculated test statistic is 0.67. Assume the conditions are satisfied.

a) Write the hypotheses to test the consumer group's claim.

b) Give the critical value if testing at the 10% level of significance.

c) Give your decision and briefly explain your reasoning.

d) Give your conclusion with respect to the consumer group's claim.

e) Explain a Type II error in the context of this question.

4- A researcher was interested in the relationship between expenditure onimports($billions) and personalincome($000s) for residents in a country. EXCEL was used to create the following simple linear regression output.

Imports and Income Excel Output:

SUMMARY OUTPUT Regression Statistics Multiple R 0.850 R Square 0.723 Observations 100 ANOVA df SS MS F Regression 130904.793 130904.793 263.121 Residual 98 8955.157 497.509 Total 99 139859.95 Coefficients Standard Error t Stat P-value Intercept -110.771 32.067 -3.454 0.000 Income 0.215 0.001 2.150 0.000