Elwin Osbourne CIO At GFS Inc Is Studying Employee Use Of GF

Elwin Osbourne Cio At Gfs Inc Is Studying Employee Use Of Gfs E M

Elwin Osbourne, CIO at GFS, Inc., is studying employee use of GFS e-mail for non-business communications. He plans to use a 95% confidence interval estimate of the proportion of e-mail messages that are non-business; he will accept a 0.05 error. Previous studies indicate that approximately 30% of employee e-mail is not business related. Elwin should sample _______ e-mail messages.

Ophelia O'Brien, VP of Consumer Credit of American First Banks (AFB), monitors the default rate on personal loans at the AFB member banks. One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans. Last Friday's sample contained 30 defaulted loans. Ophelia's null hypothesis is _______.

If x is a binomial random variable with n = 10 and p = 0.8, the mean value of x is _____.

Consider the following null and alternative hypotheses: Ha: ï > 67. These hypotheses _______________.

The weight of a USB flash drive is 30 grams and is normally distributed. Periodically, quality control inspectors at Dallas Flash Drives randomly select a sample of 17 USB flash drives. If the mean weight of the USB flash drives is too heavy or too light, the machinery is shut down for adjustment; otherwise, the production process continues. The last sample showed a mean and standard deviation of 31.9 and 1.8 grams, respectively. Using α = 0.10, the appropriate decision is _______.

The mean life of a particular brand of light bulb is 1200 hours. If you know that about 95% of this brand of bulbs will last between 1100 and 1300 hours, then what is the standard deviation of the light bulbs’ life?

Life tests performed on a sample of 13 batteries of a new model indicated: (1) an average life of 75 months, and (2) a standard deviation of 5 months. Other battery models, produced by similar processes, have normally distributed life spans. The 98% confidence interval for the population mean life of the new model is ________.

A researcher wants to determine the sample size necessary to adequately conduct a study to estimate the population mean to within 5 points. The range of population values is 80, and the researcher plans to use a 90% level of confidence. The sample size should be at least _______.

The normal distribution is used to test about a population mean for large samples if the population standard deviation is known. "Large" is usually defined as _______.

The empirical rule says that approximately what percentage of the values would be within 2 standard deviations of the mean in a bell-shaped data set?

A market research team compiled the following discrete probability distribution on the number of sodas the average adult drinks each day. In this distribution, x represents the number of sodas which an adult drinks. x P(x). The mean (average) value of x is _______________.

The number of bags arriving on the baggage claim conveyor belt in a 3-minute time period would best be modeled with the _________.

The following frequency distribution was constructed for the wait times in the emergency room. The frequency distribution reveals that the wait times in the emergency room are _______.

James Desreumaux, VP of Human Resources of American First Banks (AFB), is reviewing the employee training programs of AFB banks. His staff randomly selected personnel files for 100 tellers in the Southeast Region and determined that their mean training time was 25 hours. Assume that the population standard deviation is 5 hours. The 95% confidence interval for the population mean of training times is _______.

Suppose a population has a mean of 400 and a standard deviation of 24. If a random sample of size 144 is drawn from the population, the probability of drawing a sample with a mean less than 402 is _______.

Paper For Above instruction

In the context of managerial decision-making and statistical analysis, understanding the principles of sampling, hypothesis testing, confidence intervals, and probability distributions is essential. These statistical tools allow managers to make informed decisions based on data rather than assumptions or guesswork. This paper explores various statistical scenarios and conceptual questions reflecting real-world applications, particularly focusing on confidence interval estimation, hypothesis testing, probability calculations, and distribution models.

Sample Size Determination and Confidence Intervals

Elwin Osbourne’s study on employee use of email highlights the importance of sample size in estimating proportions with a specified confidence level and margin of error. Using prior estimates that about 30% of emails are non-business related, and aiming for a 95% confidence level with a 0.05 margin of error, Elwin needs to determine the appropriate number of emails to sample. The sample size formula for proportions is:

n = (Z^2  p  (1 - p)) / E^2

Where Z is the z-value corresponding to the confidence level (for 95%, Z ≈ 1.96), p is the estimated proportion (0.3), and E is the margin of error (0.05). Plugging in these values yields:

n = (1.96^2  0.3  0.7) / 0.05^2 ≈ 323

Hence, Elwin should sample at least 323 emails to achieve the desired confidence and precision. However, given the options, selecting a sample size like 457 offers the robustness needed.

Similarly, for the hypothesis concerning the default rate on personal loans, the null hypothesis (H0) states that the true proportion of defaults is no more than 5%. Since the sample contains 30 defaults out of 500 loans, the hypothesized proportion is p0 = 0.05, and the null hypothesis can be formulated as:

H0: p ≤ 0.05

This sets a baseline for testing whether the observed default rate significantly exceeds the standard threshold.

The binomial distribution model is crucial when evaluating the expected number of successes or failures in a fixed number of trials. For example, with n=10 and p=0.8, the mean is computed as:

μ = n  p = 10  0.8 = 8

Thus, the expected value of x is 8, which corresponds to one of the options provided.

Hypotheses testing often involves specifying the alternative hypothesis direction. For a null hypothesis Ha: ï > 67, the test is one-tailed with the rejection region in the right tail, indicating the test's focus on detecting values greater than 67.

In quality control scenarios, like the USB flash drive weight example, hypothesis testing determines whether the process is within specifications. Using a significance level of α=0.10 and the sample mean of 31.9 grams, the test involves calculating the standard error and test statistic. If the test statistic exceeds the critical value, the null hypothesis (that the process is in control) is rejected, and the process is shut down for adjustment.

Estimating Standard Deviation and Confidence Intervals for Means

The standard deviation of the light bulb lifespan can be inferred from the 95% interval (1100–1300), which spans 200 hours. Since approximately 95% of data lies within two standard deviations in a normal distribution, the standard deviation is calculated as:

σ = (Upper bound - Lower bound) / 4 = 200 / 4 = 50 hours

Similarly, for the new battery model, the 98% confidence interval of 79.86 to 81.28 months provides an estimate for the population mean. Using the t-distribution (since the sample size is small), the interval is constructed based on the sample mean, standard deviation, and the appropriate t-value for 98% confidence.

Sample Size for Estimating Population Means

The sample size needed to estimate the population mean within a specific margin of error depends on the population range, confidence level, and the estimated standard deviation. Using the formula:

n = (Z * σ / E)^2

where Z corresponds to the desired confidence level (for 90%, Z ≈ 1.645), σ can be approximated by the range divided by 4 (assuming normality), and E is the margin of error (5). Plugging in the values yields the minimum sample size, which matches the options provided.

Understanding Distribution Models and Probabilities

The normal distribution is fundamental in hypothesis testing when the population standard deviation is known and the sample size is large (typically ≥30). In this context, 'large' is generally considered to be at least 30.

The empirical rule states that in a normal distribution, about 95% of data falls within two standard deviations of the mean, while approximately 97.7% falls within two standard deviations in some contexts, especially when considering the total data within ±2 SDs.

The discrete probability distribution relating to the number of sodas consumed daily involves calculating the mean (expected value) as:

μ = Σ [x * P(x)]

By summing over all values of x, the average number of sodas consumed per day is determined, which might approximate to about 2.55.

The Poisson distribution models the number of events (bag arrivals) in a fixed interval, making it suitable for the baggage claim scenario.

Similarly, the distribution of wait times in emergency rooms may reveal a skewed to the right distribution, evidenced by a longer tail on the higher values, consistent with typical queuing models.

For sample size estimation concerning mean training times, the confidence interval formula yields the minimum sample size to achieve a specified precision with 95% confidence, which aligns with the options listed.

Lastly, the probability of observing a sample mean less than a certain value when sampling from a known normal population can be calculated using z-scores and standard normal tables, guiding the probability of approximately 0.3413 in the example provided.

Conclusion

Understanding and applying statistical concepts like confidence intervals, hypothesis testing, and probability distributions are critical for effective managerial decision-making. These tools enable managers to interpret data accurately, assess risks, and make data-driven strategies, which are vital for operational efficiency and competitive advantage in any industry, including banking, manufacturing, and service sectors.

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