Question

Suppose that a quality characteristic is normally distributed with specifications at 100 ± 20. The

process standard deviation is 6.
(a) Suppose that the process mean is 100. Calculate P and P.
(b) Suppose that the process mean is 106. Calculate P and P.
Q2. Suppose that a process has P = 1.50 and P = 1.00 with = 103 and = 2. Moreover, we
know that the process shifts to the right of the target dimension. Calculate both the upper specification
limit and the lower specification limit.
Q3. Each item produced by a certain manufacturer is, independently, of acceptable quality with
probability 0.95. Approximate the probability that at most 10 of the next 150 items produced are
unacceptable.
Q4. During a local election between two candidates, exit polls based on a sample of 400 voters indicated
that 54% of the voters supported the incumbent candidate. Construct a 90% confidence interval for the
percentage of votes that the incumbent has received in this election.

Answer 1

Statistical methods allow us to calculate z-scores, use the Central Limit Theorem, and construct confidence intervals. For example, Susan's exam score is 2 standard deviations above the mean, showing outstanding performance. Confidence intervals estimate the true population proportion with a certain confidence level.

Step-by-step explanation:

Understanding Standard Deviation and Probability

To address the questions involving standard deviation and probability, we must apply various statistical concepts, including z-scores, the Central Limit Theorem, the empirical rule, and confidence intervals. These concepts help interpret data in relation to the mean and standard deviation, calculate probabilities of events, and construct confidence intervals for population parameters based on sample statistics.

Z-Score Calculation

For example, if Susan scored a 95 on her biology final exam and the class scores were normally distributed with a mean of 85 and a standard deviation of 5, we can calculate her z-score. The z-score is obtained by subtracting the mean from Susan's score and then dividing by the standard deviation, resulting in a z-score of (95-85)/5=2. This means her score is 2 standard deviations above the mean, indicating she performed exceptionally well compared to her classmates.

Central Limit Theorem

The Central Limit Theorem states that for a sufficiently large sample size, the distribution of the sample mean will tend to be normally distributed, regardless of the population's distribution. This allows us to make inferences about the population mean based on the sample mean.

Confidence Intervals

To construct a 90% confidence interval for the percentage of votes the incumbent candidate received, we use the sample proportion (0.54), the standard error, and the critical z-value (1.645 for a 90% confidence level). This interval will give us a range within which we can be 90% confident that the true proportion of votes for the incumbent lies.