Question

Continuous or Discrete? Identify each variable as continuous or discrete: a. Number of homicides in Detroit during a one-month period b. Length of time between arrivals at an outpatient clinic c. Number of typing errors on a page of manuscript d. Number of defective lightbulbs in a package containing four bulbs e. Time required to finish an examination

Short answer

Question: Determine whether each of the following variables is continuous or discrete: a. Number of homicides in Detroit during a one-month period b. Length of time between arrivals at an outpatient clinic c. Number of typing errors on a page of manuscript d. Number of defective lightbulbs in a package containing four bulbs e. Time required to finish an examination

Step-by-step solution

a. Number of homicides in Detroit during a one-month period

This variable is discrete, as the number of homicides can only be expressed as whole numbers (it doesn't make sense to say that there were 2.5 homicides, for example). Also, the number of possible values is limited: the least amount of homicides is 0, and there is a maximum possible number of homicides that could happen in a month.

b. Length of time between arrivals at an outpatient clinic

This variable is continuous, as the time between arrivals can vary continuously, and it can take any value within an interval (for example, there could be any real number value between two specific arrival times, such as 2.23 minutes, 2.24 minutes, or any other time between these two arrivals].

c. Number of typing errors on a page of manuscript

This variable is discrete, because the number of typing errors must be a whole number (you cannot have half of an error). Additionally, there is a limited number of possible values for this variable, from 0 errors up to the number of characters on the manuscript page.

d. Number of defective lightbulbs in a package containing four bulbs

This variable is discrete, since the number of defective lightbulbs can only be expressed as a whole number (e.g., 1, 2, 3, or 4), indicating the specific count of defective lightbulbs in the package. The number of possible values is also limited, as it must be between 0 and 4.

e. Time required to finish an examination

This variable is continuous, as the time required to complete an examination can take any real value within a given interval, such as a student taking 24.5 minutes to finish the exam, another taking 38.2 minutes, and so on. The time cannot be restricted to only specific values (discrete), and it varies continuously within a range.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions using data from a study or an experiment. It involves making an initial assumption, known as the null hypothesis \( H_0 \), and an alternative hypothesis \( H_a \). For example, in the context of the exercise we could have a null hypothesis stating that there is no difference in the mean number of homicides during a one-month period when comparing two different years.

The process requires choosing a significance level, \( \alpha \), and then calculating a test statistic from the sample data. The test statistic is compared against a critical value determined by the \( \alpha \), and the distribution of the test under the null hypothesis. If the test statistic is more extreme than the critical value, we reject the null hypothesis in favor of the alternative.

The context of discrete variable like the number of homicides can be analyzed using certain types of hypothesis tests, such as a chi-squared test for goodness of fit, which can assess if the observed frequencies differ from expected frequencies for certain categories.

It's very important to clearly define the null and alternative hypotheses, ensure appropriate conditions for the hypothesis test are met, and choose the right test for the type of data, whether discrete or continuous.

Descriptive Statistics

Descriptive statistics are used to summarize or describe features of a dataset. They can provide simple summaries about the sample and the measures. These include measures of central tendency like the mean, median, and mode, as well as measures of variability like the range, variance, and standard deviation.

In the exercise we looked at different variables like the number of typing errors on a page of manuscript, which is a discrete variable. Descriptive statistics for discrete variables will often include the frequency and percentage of each value, perhaps displayed in a frequency table.

• For instance, if we collected data from multiple pages of manuscript, we could use a frequency chart to show how many times each number of errors occurred.
• For continuous variables such as 'time required to finish an examination', we might calculate the average time, as well as the standard deviation which would tell us how much variation there is in the times.

Descriptive statistics provide powerful insights into data, making it easier to communicate numerical information and identify patterns or outliers, especially in large datasets.

Probability Distributions

Probability distributions describe how the values of a random variable are distributed. They are categorized as either discrete or continuous, depending on the type of variable they are used to model.

A discrete probability distribution is relevant for the outcomes of discrete variables, such as the number of defective lightbulbs in a package. The probabilities of discrete distributions sum to one and can be listed out in a table or represented graphically with a probability mass function (PMF). Common discrete distributions include the binomial distribution and Poisson distribution.

For continuous variables, such as time between arrivals at an outpatient clinic, the probability distribution is described using a probability density function (PDF). Here, the probabilities are calculated over intervals, not specific points, and one common example is the normal distribution.

Understanding the appropriate distribution allows us to make predictions about data and compute probabilities for different outcomes, which in turn assists in areas such as predictive modeling and risk assessment.