Exam 7: Sampling Methods and the Central Limit Theorem

A study of a company's practice regarding the payment of invoices revealed that on the average an invoice was paid 20 days after it was received. The standard deviation equalled five days. A sample of 25 invoices is selected. Assuming that the distribution is normal, what percent of the sampled invoices were paid within 15 days of receipt?

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples 50 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

i. If the sample size keeps getting larger and larger and finally equals the size of the population, there would be no error in predicting the population mean because the sample size and the size of the population would be the same. ii. A simple random sample assumes that each item or person in the population has an equal chance of being included. iii. We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error.

i. If a population is not normally distributed, the sampling distribution of the sample means tends to approximate a normal distribution. ii. The Central Limit Theorem states that if the sample size n is sufficiently large, the sampling distribution of the means will be approximately normal no matter whether the population is normally distributed, skewed, or uniform. iii. Based on the sampling distribution of the means and the central limit theorem, the sample mean can be used as a good estimator of the population mean, assuming that the size of the sample is sufficiently large.

A retailer claims that 90% of its customers are "pleased" or "very pleased" with the customer service. In a survey of 300 customers taken last week, what is the probability that between 80% and 90% will be "pleased" or "very pleased" with the service?

(i. The standard error of the mean is the standard deviation of the sampling distribution of the sample means. (ii) The standard deviation of the sampling distribution of the mean is always smaller than the standard deviation of the population under study. (iii) For a sampling distribution of the means, 90% of the means would be between $\pm$ 1.96 standard deviations.

Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as Class A large; those from $50 to $100 million as Class A medium size; and those between $25 and $50 million, and so on. Samples were then selected from each of these groups. What is this type of sampling called?

(i. As the sample size (n) increases, the spread in the distribution of the sample means increases. (ii) If the sampling size equals the population size, the sampling error is zero. (iii) If a population is normally distributed, the sampling distribution of the mean is normally distributed.

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes. What is the probability that the mean completion time will be more than 120 minutes?

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples 40 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A retailer claims that 90% of its customers are "pleased" or "very pleased" with the customer service. In a survey of 300 customers taken last week, what is the probability that 84% or more will be "pleased" or "very pleased" with the service?

i. An estimate of the population mean based on a large sample is less reliable than an estimate made using a small sample. ii. The standard error of the mean will vary according to the size of the sample that is in the denominator. As the sample size n gets larger, the variability of the sample means gets smaller. iii. To determine the value of the standard error of the mean, the total error is divided by the sample size.

(i. As the sample size (n) increases, the spread in the distribution of the sample means stays the same. (ii) If the sampling size equals the population size, the sampling error is 1. (iii) If a population is normally distributed, the sampling distribution of the mean is normally distributed.

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples 60 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

Sampling error is the difference between a corresponding sample statistic and the

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 3 or more parts out of a sample of 60 to be defective?

An experiment involves randomly selecting a sample of 256 middle managers for study. One item of interest is their mean annual income. The sample mean is computed to be $35,420 and the sample standard deviation is $2,050. What is the standard error of the mean?

i. It is often not feasible to study the entire population because it is impossible to check all the items in the population. ii. Sampling a population is often necessary because the cost of studying all the items in the population is prohibitive. iii. Sampling is a sign of laziness on the part of the statistician.

(i. For populations scattered in a wide area, the preferred technique for sampling is cluster sampling. (ii) If the population can be divided into homogeneous subgroups, stratified random sampling is the best sampling method to use. (iii) If every k-th item in the population sequence is selected, you are using cluster sampling.

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5% underweight. The foreman samples 50 bags with each shipment and if the bags average more than 5% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.