Problem 3 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

The numbers of text messages sent each day by a random sample of 30 teen cellphone users are shown in the table. Estimate the population mean \(\mu\). Number of Text Messages $$ \begin{array}{|l|l|l|l|l|} \hline 30 & 60 & 59 & 83 & 41 \\ 37 & 66 & 63 & 60 & 92 \\ 53 & 42 & 47 & 32 & 79 \\ 53 & 80 & 41 & 51 & 85 \\ 73 & 71 & 69 & 31 & 69 \\ 57 & 60 & 70 & 91 & 67 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The estimated population mean is the result obtained in Step 3. Calculation in brief is: sum all messages and divide by the total number of observations (30).

Step 1 - Addition

Start by adding up all the values given in the table. This represents the total number of text messages sent within this sample. So, add: \(30+60+59+83+41+37+66+63+60+92+53+42+47+32+79+53+80+41+51+85+73+71+69+31+69+57+60+70+91+67\)

Step 2 - Compute the Mean

Next, calculate the mean by dividing the total found in the first step by the total number of observations, which in this case is 30.

Step 3 - Result

The result you obtain from Step 2 is the estimated population mean (\(\mu\)) of the number of text messages sent per day.

Key Concepts

Statistical Mean

The statistical mean, commonly known as the average, is a crucial concept in data analysis. It is the sum of all observations in a dataset divided by the number of observations. In practical terms, it provides a central value around which the data points can be expected to cluster. For example, when examining the daily text messages sent by teens, the mean gives us an insight into the typical usage patterns across the sample.

In order to calculate the mean from our sample data, we add together all the individual message counts and then divide by the total number of counts. This method can be represented by: \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \] In the context of our exercise, we would sum up all text messages counted in the sample and divide by 30, the size of the sample.

Sample Data Analysis

Sample data analysis involves evaluating a subset of a larger population to estimate characteristics of the whole population. While it offers a practical method for studying large populations, sample analysis can only provide estimates, not precise figures. Ensuring the sample is random and representative is important to obtain reliable results.

The data analysis process usually starts with data collection, followed by summarizing the data through various measurements such as mean, median, variance, etc. The mean is often chosen for its simplicity and informative nature as it reflects the central tendency of the data. In our exercise, by analyzing the sample of teen text message data, we aim to estimate the average number of text messages a teen sends in a day across the entire population.

Data Collection in Statistics

Data collection is a foundational step in the field of statistics that involves gathering information to be used for statistical analysis. The quality, accuracy, and comprehensiveness of this data directly affect the reliability of the statistical conclusions.

Data can be collected through various methods such as surveys, experiments, or observing and recording information. For our exercise on estimating the mean number of text messages sent by teens, data was collected from a random sample, which helps to reduce bias and improve the estimate's accuracy. Ensuring the sample is representative of the broader population is essential; otherwise, the mean calculated from the sample may not effectively estimate the population mean.