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Chapter 9: Problem 2

A survey of 100 random full-time students at a large university showed the mean number of semester units that students were enrolled in was \(15.2\) with a standard deviation of \(1.5\) units. a. Are these numbers statistics or parameters? Explain. b. Label both numbers with their appropriate symbol (such as \(\bar{x}, \mu, s\), or \(\sigma)\).

Short Answer

Expert verified
a. Both numbers refer to statistics since they have been calculated using a sample, not the entire population.\n b. The mean number of semester units is labeled as \(\bar{x}=15.2\) and the standard deviation is labeled as \(s=1.5\).

Step by step solution

01

Identify Statistics or Parameters

The mean number of semester units is calculated from a sample of 100 students from a large university, so it is a statistic, not a parameter. Similarly, the standard deviation is also computed from this sample, making it a statistic as well.
02

Label with Appropriate Symbol

The mean value calculated from a sample is represented with the symbol \(\bar{x}\). Thus, the mean number of semester units should be referred to as \(\bar{x}=15.2\). The standard deviation from a sample is represented by the symbol 's', so the standard deviation of semester units will be \(s=1.5\).

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Key Concepts

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

Statistics vs Parameters
To understand the difference between statistics and parameters, first, think about the scope from which data is collected.
Statistics are numerical characteristics derived from a sample, or a portion of the population. In contrast, parameters are numerical characteristics describing an entire population. Consider this exercise, where the mean number of semester units and standard deviation are calculated based on a sample of 100 students.
Since these values are derived from a subset of the student population, they are considered statistics, not parameters. This distinction is crucial in data analysis, as it helps determine the generalizability of the findings. In practice, parameters represent fixed values, while statistics are variables that can change depending on the chosen sample.
Often, researchers rely on statistics to make inferences about parameters when studying populations.
Sample Mean
The sample mean is an essential concept when working with statistical data.
It gives us a central value or "average" derived from a sample, helping to summarize and describe data points.Mathematically, the sample mean is denoted by the symbol \(\bar{x}\). You calculate it by adding up all the observed values in your sample and then dividing by the number of observations.For the exercise in question, the sample mean tells us that the average number of semester units students are enrolled in is \(15.2\).
This piece of information provides an insight into student enrollment patterns and can guide university resource allocation and scheduling decisions.
Sample Standard Deviation
Sample standard deviation, denoted by the symbol \(s\), measures how spread out the values in a data set are.
It indicates the average deviation of each data point from the sample mean.To compute the sample standard deviation:
  • Subtract the sample mean from each data point and square the result.
  • Sum all these squared differences.
  • Divide this sum by one less than the number of observations in the sample (\(n - 1\)).
  • Finally, take the square root of this value to get the standard deviation.
For the exercise, a standard deviation of \(1.5\) units tells us that on average, student enrollment per semester fluctuates by about 1.5 units around the mean.
Understanding this variability helps with planning for student support services and managing course demands.

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Most popular questions from this chapter

A McDonald's fact sheet says its cones should weigh \(3.18\) ounces (converted from grams). Suppose you take a random sample of four cones, and the weights are \(4.2,3.4,3.9\), and \(4.4\) ounces. Assume that the population distribution is Normal, and, for all three parts, report the alternative hypothesis, the \(t\) -value, the p-value, and your conclusion. The null hypothesis in all three cases is that the population mean is \(3.18\) ounces. a. Test the hypothesis that the cones do not have a population mean of 3\. 18 ounces. b. Test the hypothesis that the cones have a population mean less than \(3.18\) ounces. c. Test the hypothesis that the cones have a population mean greater than \(3.18\) ounces.

The weights of four randomly and independently selected bags of potatoes labeled 20 pounds were found to be \(21.0\), \(21.5,20.5\), and \(21.2\) pounds. Assume Normality. a. Find a \(95 \%\) confidence interval for the mean weight of all bags of potatoes. b. Does the interval capture \(20.0\) pounds? Is there enough evidence to reject a mean weight of 20 pounds?

State whether each of the following changes would make a confidence interval wider or narrower. (Assume that nothing else changes.) a. Changing from a \(90 \%\) confidence level to a \(99 \%\) confidence level b. Changing from a sample size of 30 to a sample size of 200 c. Changing from a standard deviation of 20 pounds to a standard deviation of 25 pounds

The weights of four randomly chosen bags of horse carrots, each bag labeled 20 pounds, were \(20.5,19.8,20.8\), and \(20.0\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of horse carrots. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. \(95 \%\) of all sample means based on samples of the same size will be between _____ and _____. ii. I am \(95 \%\) confident that the population mean is between _____ and _____. iii. We are \(95 \%\) confident that the boundaries are _____ and _____. b. Can you reject a population mean of 20 pounds? Explain.

According to a 2018 Money magazine article, Maryland has one of the highest per capita incomes in the United States, with an average income of $$\$ 75,847$$. Suppose the standard deviation is $$\$ 32,000$$ and the distribution is right-skewed. A random sample of 100 Maryland residents is taken. a. Is the sample size large enough to use the Central Limit Theorem for means? Explain. b. What would the mean and standard error for the sampling distribution? c. What is the probability that the sample mean will be more than $$\$ 3200$$ away from the population mean?
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