High School

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**Question 1:**

Suppose the measures to ensure that all manufactured motor vehicles meet requirements with regards to carbon monoxide emission indicate that the standard deviation per unit volume of exhaust fumes must not exceed 0.03 percent carbon monoxide emission particles per unit volume. A random sample of ten newly manufactured motor vehicles is tested, and the following carbon monoxide emissions are recorded: 0.23, 0.27, 0.31, 0.25, 0.30, 0.29, 0.32, 0.26, 0.24, 0.22.

a) Calculate the point estimate of the standard deviation of the carbon monoxide emissions. (10 points)

b) What is the 99% confidence interval estimate for the population variance of the carbon monoxide emissions? (10 points)

c) Provide an interpretation for the answer in part (b). (5 points)

d) Suppose the level of confidence is reduced. What will happen to the answer in part (b)? Explain. (5 points)

e) Use the critical value approach and test a hypothesis at a 5% level of significance to determine whether the requirements are met. Show all hypothesis test steps. (10 points)

**Question 2:**

Suppose you are assigned a task to investigate the relationship between the selling price and the valuation of plots sold by a local municipality. Data was obtained for a random sample of ten plots.

| Plot | Selling Price (N$ 000) | Valuation (N$ 000) |
|------|------------------------|--------------------|
| 1 | 120 | 72 |
| 2 | 100 | 68 |
| 3 | 140 | 72 |
| 4 | 150 | 70 |
| 5 | 155 | 75 |
| 6 | 100 | 50 |
| 7 | 150 | 58 |
| 8 | 200 | 90 |
| 9 | 80 | 56 |
| 10 | 145 | 70 |

a) Discuss the difference between correlation analysis and regression analysis. (10 points)

b) Use the method of least squares to estimate the regression equation between selling price and valuation. (15 points)

c) Provide an interpretation for the slope coefficient. (5 points)

d) Use the estimated regression equation to make a prediction for the value of the dependent variable when the independent variable is N$ 85,000. (5 points)

e) Calculate and interpret the coefficient of determination. (10 points)

f) Calculate and interpret the correlation coefficient. (5 points)

**Question 3:**

Consider the basic food items consumed by a rural household:

| Food Item | Consumption (June 2021) | Consumption (July 2021) | Price (N$) June 2021 | Price (N$) July 2021 |
|-------------|-------------------------|-------------------------|----------------------|----------------------|
| Bread (loaves) | 25 | 26 | 10.30 | 10.70 |
| Cooking oil (liters) | 5 | 6 | 45.00 | 59.40 |
| Rice (kg) | 10 | 15 | 34.50 | 36.20 |
| Meat (kg) | 8 | 16 | 88.90 | 92.20 |

a) Which food item has had the lowest relative quantity of consumption increase between June and July? (3 points)

b) Which food item has had the lowest relative price increase between June and July 2021? (3 points)

c) Calculate and interpret a Paasche quantity index for July 2021. (4 points)

Answer :

a) The point estimate of the standard deviation is approximately 0.037.

b) The 99% confidence interval estimate for the population variance is (0.000034, 0.002505).

c) The 99% confidence interval suggests that we can be 99% confident that the true population variance of carbon monoxide emissions lies within the calculated interval.

d) If the level of confidence is reduced, the width of the confidence interval will decrease, becoming narrower, as it will require less certainty to capture the true population variance.

e) Hypothesis Test Steps:

Null Hypothesis (H0): The population variance is ≤ 0.03.

Alternative Hypothesis (Ha): The population variance is > 0.03.

Test Statistic: χ² = (n - 1) * s² / σ² ~ χ² with (n - 1) degrees of freedom.

Significance Level: α = 0.05.

Critical Value: χ²_critical = χ²_(1 - α, n-1) = χ²_(0.95, 9).

Calculate the Test Statistic: χ² = (9) * (0.001609) / 0.03 = 0.48163.

Compare χ² to χ²_critical: Since 0.48163 < 16.919, fail to reject H0.

Conclusion: There is not enough evidence to reject the claim that the population variance is ≤ 0.03 at a 5% significance level.

(a) The point estimate of the standard deviation can be calculated using the sample data and the formula for the sample standard deviation.

(b) The confidence interval for the population variance can be calculated using the chi-squared distribution and the given sample data.

(c) The confidence interval interpretation means that we can be 99% confident that the true population variance lies within the calculated interval.

(d) A reduced level of confidence would result in a narrower confidence interval as it requires less certainty.

(e) The hypothesis test follows the steps of setting up null and alternative hypotheses, calculating the test statistic, comparing it to the critical value, and drawing a conclusion about the population variance based on the significance level. In this case, since the calculated test statistic is less than the critical value, we fail to reject the null hypothesis, indicating that there is not enough evidence to suggest that the population variance exceeds 0.03.

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