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Chapter 7: Problem 30

For a population, \(N=10,000, \mu=124\), and \(\sigma=18\). Find the \(z\) value for each of the following for \(n=36\). a. \(\bar{x}=128.60\) b. \(\bar{x}=119.30\) c. \(\bar{x}=116.88\) d. \(\bar{x}=132.05\)

Short Answer

Expert verified
The calculated z-scores are:a) 1.80 b) -1.65 c) -2.36 d) 2.68

Step by step solution

01

Identify known variables

From the problem statement, we identify the known variables: \(N=10,000, \mu=124, 蟽=18\) and \(n=36\). The value of \(\bar{x}\) varies for each part of the problem.
02

Calculation for a. \(\bar{x}=128.60\)

For \(\bar{x}=128.60\), apply the formula \(Z = \frac{(X-渭)}{蟽/ \sqrt{n}}\). Substitute the values into the formula, \(Z = \frac{(128.60 - 124)}{18/ \sqrt{36}}\), which simplifies to \(Z = 1.80\).
03

Calculation for b. \(\bar{x}=119.30\)

For \(\bar{x}=119.30\), apply the formula \(Z = \frac{(X-渭)}{蟽/ \sqrt{n}}\). Substitute the values into the formula, \(Z = \frac{(119.30 - 124)}{18/ \sqrt{36}}\), which simplifies to \(Z = -1.65\).
04

Calculation for c. \(\bar{x}=116.88\)

For \(\bar{x}=116.88\), apply the formula \(Z = \frac{(X-渭)}{蟽/ \sqrt{n}}\). Substitute the values into the formula, \(Z = \frac{(116.88 - 124)}{18/ \sqrt{36}}\), which simplifies to \(Z = -2.36\).
05

Calculation for d. \(\bar{x}=132.05\)

For \(\bar{x}=132.05\), apply the formula \(Z = \frac{(X-渭)}{蟽/ \sqrt{n}}\). Substitute the values into the formula, \(Z = \frac{(132.05 - 124)}{18/ \sqrt{36}}\), which simplifies to \(Z = 2.68\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean
The population mean, often denoted by the Greek letter \( \mu \), is the average of all observations in the entire population. It provides a central value and is calculated by summing all the values and dividing by the population size \( N \). For example, if we consider a population with \( \mu = 124 \), it means that this is the average value for the entire population. Understanding the population mean is crucial, as it serves as a benchmark to compare samples we might take from that population. When calculating the z-score, the population mean represents the expected value around which sample means fluctuate.
Sample Size
Sample size, represented by \( n \), is the number of observations in a sample drawn from a population. In statistical analysis, the sample size is pivotal because it affects the accuracy and reliability of inferences made about the population. A larger sample size generally means more reliable estimates, as it tends to better represent the population. In our exercise, the sample size is \( n = 36 \). This information is key when calculating the standard error, which in turn is used to find the z-score in hypothesis testing.
Standard Deviation
The standard deviation, \( \sigma \), quantifies the amount of variation or deviation in a set of data values from the mean. In simpler terms, it tells us how spread out the values are in the population. A small standard deviation indicates the data values are close to the mean, whereas a large standard deviation signifies more extensive variability. In our example, the population standard deviation is \( \sigma = 18 \). This value is a crucial component in the calculation of the standard error, which is used as the denominator in the z-score formula.
Standard Error
The standard error (SE) measures the spread or variability of the sample mean estimates around the population mean. Specifically, it is the standard deviation of the sampling distribution of the sample mean. It is computed using the formula \( \text{SE} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size. In our exercise, \( \text{SE} = \frac{18}{\sqrt{36}} = 3 \). This number reflects how much sample means deviate from the actual population mean due to randomness and is used to calculate the z-score, indicating how many standard errors a sample mean is from the population mean.

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Most popular questions from this chapter

A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard deviation of 25 pounds. If 35 of these people board the elevator, what is the probability that their combined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.

Explain briefly the meaning of nonsampling errors. Give an example. Do such errors occur only in a sample survey, or can they occur in both a sample survey and a census?

The package of Ecosmart Led 75-watt replacement bulbs that use only 14 watts claims that these bulbs have an average life of 24,966 hours. Assume that the lives of all such bulbs have an approximate normal distribution with a mean of 24,966 hours and a standard deviation of 2000 hours. Find the probability that the mean life of a random sample of 25 such bulbs is a. less than 24,400 hours b. between 24,300 and 24,700 hours c. within 650 hours of the population mean d. less than the population mean by 700 hours or more

According to a Gallup poll conducted January 5-8, 2014, 67\% of American adults were dissatisfied with the way income and wealth are distributed in America. Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 400 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

Dartmouth Distribution Warehouse makes deliveries of a large number of products to its customers. It is known that \(85 \%\) of all the orders it receives from its customers are delivered on time. Let \(\hat{p}\) be the proportion of orders in a random sample of 100 that are delivered on time. Find the probability that the value of \(\hat{p}\) will be a. between \(.81\) and \(.88\) b. less than \(.87\)
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