An experiment was conducted to determine the abrasion resistance of a new type of automobile paint. Twelve different strips of metal were painted with the new paint. The abrasion resistance of each piece was then tested on a machine. In the sample, the abrasion resistance values ranged from 2.1 units to 4.3 units.
1. What is the parameter of interest in this study?
2. What are the two assumptions required for constructing a confidence interval for the population mean in this case?
3. The Q-Q plot for these data is shown. What assumption is this plot used to assess? Based on this plot, do we have evidence that the assumption holds? Why or why not?
4. The sample mean resistance is 3.2 and the standard deviation is 0.73. Which distribution will be used to construct a 95% CI for the population mean? BE SPECIFIC
5. The sample mean resistance is 3.2 and the standard deviation is 0.73. What is the 95% CI for the population mean resistance value?
6. Which of the following is the correct interpretation of the 95% confidence level?(
1. The parameter of interest in this study is the average abrasion resistance of the new type of automobile paint for the population of all painted strips of metal.
2. The two assumptions required for constructing a confidence interval for the population mean in this case are:
a) The data follows a normal distribution.
b) The samples are representative of the population.
3. The Q-Q plot is used to assess the assumption of normality. Based on the plot, if the data points closely follow the diagonal line, it suggests that the assumption of normality holds. If the points deviate significantly from the diagonal line, it indicates a departure from normality. Without seeing the plot, it's not possible to determine whether the assumption of normality holds or not.
4. Since the sample size is greater than 30 and we have the population standard deviation, we will use the normal distribution to construct a 95% confidence interval for the population mean.
5. Using the sample mean resistance of 3.2 and the standard deviation of 0.73, and assuming the data follows a normal distribution, the 95% confidence interval for the population mean resistance value can be calculated as follows:
Lower Limit = Sample Mean - (Z-value * Standard Error)
Upper Limit = Sample Mean + (Z-value * Standard Error)
Assuming a 95% confidence level, the Z-value corresponding to a two-tailed test is approximately 1.96 (obtained from the standard normal distribution table). The standard error is calculated as the sample standard deviation divided by the square root of the sample size.
Therefore, the 95% confidence interval for the population mean resistance value is:
Lower Limit = 3.2 - (1.96 * (0.73 / sqrt(12)))
Upper Limit = 3.2 + (1.96 * (0.73 / sqrt(12)))
6. The correct interpretation of the 95% confidence level is that if we were to repeat this study 100 times and construct a 95% confidence interval for the population mean each time, approximately 95 of those intervals would capture the true population mean value. In other words, we can be 95% confident that the true population mean falls within the calculated confidence interval.
1. The parameter of interest in this study is the population mean abrasion resistance of the new type of automobile paint.
2. The two assumptions required for constructing a confidence interval for the population mean in this case are:
a) The data follows a normal distribution.
b) The variability of the population can be reasonably represented by the sample standard deviation.
3. The Q-Q plot is used to assess the assumption of normality in the data. Based on this plot, we can visually examine whether the data points fall approximately along a straight line, indicating a normal distribution. If the points deviate significantly from a straight line, then there might be evidence that the assumption of normality does not hold. Without the actual Q-Q plot, it is not possible to determine if the assumption holds or not.
4. Since the sample mean and standard deviation are known, and the sample size is sufficiently large (assuming a sample size of at least 30), a 95% confidence interval for the population mean can be constructed using the z-distribution.
5. To construct a 95% CI for the population mean resistance value, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
Given the sample mean resistance of 3.2, standard deviation of 0.73, and assuming a sample size of at least 30, we can use the 1.96 critical value from the z-table for a 95% confidence level. Substituting these values into the formula, we can calculate the confidence interval.
6. The correct interpretation of the 95% confidence level is that if we were to repeat the sampling process multiple times and construct a confidence interval each time, approximately 95% of those intervals would contain the true population mean resistance value. This means that we can be reasonably confident (with 95% certainty) that the interval we calculated contains the true population mean.