statistics - chapter one and two test - fall 2023
four questions 1-5, describe what each term means in statistics. do not recite formal
definitions, rather explain what it means as if to someone who is new to the class. use
examples if you like.
1. population ...
2. sample ...
3. qualitative data ...
4. random sample ...
5. biased data ...
6. why are samples often used in statistical surveys instead of a population?
...
if a sample is used, why is it so important that it be
epresentative\?
...
is it possible for 80% of the numbers in a data set to be \above average\? explain.
...
what type of graph(s) are appropriate for qualitative data? for quantitative data?
...

Question

Accepted Answer

To address these statistics - related questions, we'll go through each one:

### Question 1: Define "Population" in Statistics
# Brief Explanations:
In statistics, a population is the entire set of individuals, objects, or events that we want to study and draw conclusions about. For example, if we want to study the average height of all high - school students in a country, the population would be every single high - school student in that country.
# Answer:
A population is the complete set of all individuals, objects, or events under study.

### Question 2: Define "Sample" in Statistics
# Brief Explanations:
A sample is a subset of the population. It is a smaller group that is selected from the population. Using the previous example of high - school students' height, if we select 1000 high - school students from different schools in the country to measure their heights, these 1000 students form a sample.
# Answer:
A sample is a subset of the population, selected for study.

### Question 3: Define "Qualitative Data" in Statistics
# Brief Explanations:
Qualitative data is non - numerical data that is used to describe characteristics or qualities. It can be in the form of categories, labels, or descriptions. For example, the colors of cars in a parking lot (red, blue, green etc.) or the types of animals in a zoo (mammals, birds, reptiles) are qualitative data.
# Answer:
Qualitative data is non - numerical data that describes qualities or characteristics (e.g., colors, types, opinions).

### Question 4: Define "Random Sample" in Statistics
# Brief Explanations:
A random sample is a sample where each member of the population has an equal chance of being selected. For instance, if we want to randomly sample 50 students from a school with 1000 students, we can assign a number to each student and then use a random number generator to select 50 numbers. Each student has the same probability of being chosen.
# Answer:
A random sample is a sample where each population member has an equal chance of selection.

### Question 5: Define "Biased Data" in Statistics
# Brief Explanations:
Biased data is data that is systematically skewed or distorted. This can happen due to a flawed sampling method, measurement error, or other factors. For example, if we want to survey the average income of people in a city but only sample people from high - end neighborhoods, the data we collect will be biased because it does not represent the entire city's population.
# Answer:
Biased data is data that is systematically skewed (e.g., due to flawed sampling/measurement).

### Question 6: Why are samples often used in Statistical surveys instead of a population?
# Brief Explanations:
- **Cost**: Studying an entire population can be extremely expensive. For example, if we want to study the quality of all the light bulbs produced by a factory in a year (which could be in the millions), it would cost a lot of money to test each and every bulb.
- **Time**: It would take a very long time to collect data from an entire population. If we are doing a survey on the reading habits of all the citizens in a large country, it would be time - consuming to reach every single citizen.
- **Feasibility**: In some cases, it is impossible to study the entire population. For example, if we want to study the lifespan of all the stars in the universe, we can't observe each star until it dies.
# Answer:
Samples are used because studying an entire population is often too costly, time - consuming, or infeasible.

### Question 7: If a sample is used, why is it so important that it be "representative"?
# Brief Explanations:
A representative sample accurately reflects the characteristics of the population. If the sample is not representative, the conclusions we draw from the sample will not be valid for the population. For example, if we want to study the voting preferences of a city's population but our sample only includes people from one political party's stronghold, the results of our survey will not accurately represent the city's overall voting preferences.
# Answer:
A representative sample ensures that conclusions about the sample can be generalized to the population (invalid conclusions result from non - representative samples).

### Question 8: Is it possible for 80% of the numbers in a data set to be "above average"? Explain.
# Brief Explanations:
Yes, it is possible. The average (mean) is affected by extreme values. For example, consider the data set: 1, 1, 1, 1, 5. The mean is $\frac{1 + 1+1 + 1+5}{5}=\frac{9}{5} = 1.8$. Now, let's count the number of values above the mean. The values are 1, 1, 1, 1, 5. The values above 1.8 are just 5, which is 20% of the data. But if we have a data set like: 1, 1, 1, 1, 100. The mean is $\frac{1+1 + 1+1+100}{5}=\frac{104}{5}=20.8$. The values above the mean are just 100, which is 20% of the data. Wait, maybe a better example: consider the data set with values: 1, 2, 3, 4, 100. The mean is $\frac{1 + 2+3 + 4+100}{5}=\frac{110}{5} = 22$. The values above 22 are only 100 (20% of the data). Wait, perhaps a data set with a skewed distribution. Let's take the data set: 1, 1, 1, 1, 1, 1, 1, 1, 1, 10. The mean is $\frac{9	imes1+10}{10}=\frac{19}{10}=1.9$. The number of values above 1.9: the value 10 is above 1.9, and the other 9 values are 1 which is below. So only 10% are above. Wait, maybe I made a mistake. Let's think of a data set with a mode or median. Wait, no, the mean can be pulled up by a few large values. Let's take the data set: 1, 1, 1, 1, 5. Wait, no. Wait, let's consider a data set where the mean is less than most of the values. Wait, suppose we have a data set with 10 numbers: 1, 1, 1, 1, 1, 1, 1, 1, 1, 100. The mean is (9*1 + 100)/10=(9 + 100)/10 = 10.9. Now, how many numbers are above 10.9? Only 100, which is 10% of the data. Wait, maybe a different approach. Let's use the median. Wait, no, the question is about the mean. Wait, let's take a data set with 5 numbers: 1, 1, 1, 1, 10. The mean is (4*1+10)/5 = 14/5 = 2.8. The numbers above 2.8: only 10, which is 20% of the data. Wait, maybe I am wrong. Wait, let's consider a data set where the mean is a very low value because of a few extremely low values. For example, data set: 1, 10, 10, 10, 10. The mean is (1 + 10+10 + 10+10)/5=(41)/5 = 8.2. The numbers above 8.2: 10, 10, 10, 10. That's 4 out of 5 numbers, which is 80% of the data. Ah! There we go. So in this data set, the mean is 8.2, and 4 out of 5 (80%) of the numbers (the four 10s) are above the mean. The reason this is possible is that the mean is affected by the extreme low value (1 in this case). The mean is pulled down by the low value, so most of the other values are above the mean.
# Answer:
Yes. For example, in the data set {1, 10, 10, 10, 10}, the mean is $\frac{1 + 10+10 + 10+10}{5}=8.2$. 4 out of 5 (80%) values (the 10s) are above 8.2. The mean is pulled down by the low value (1), so most values can be above the mean.

### Question 9: What type of graph(s) are appropriate for qualitative data? For quantitative data?
# Brief Explanations:
- **Qualitative Data**: Bar graphs and pie charts are appropriate. A bar graph can be used to show the frequency of each category. For example, if we have data on the favorite colors of a group of people (red, blue, green), we can use a bar graph where each bar represents a color and the height of the bar represents the number of people who chose that color. A pie chart can be used to show the proportion of each category in the whole. For example, if 30% of people like red, 40% like blue, and 30% like green, a pie chart can visually represent these proportions.
- **Quantitative Data**: Histograms (for continuous data), bar graphs (for discrete data), line graphs (to show trends over time), and box - and - whisker plots (to show the distribution of data) are appropriate. A histogram is used to show the frequency distribution of continuous data. For example, the distribution of heights of students in a class. A line graph can be used to show how a quantity changes over time, like the temperature over a week. A box - and - whisker plot shows the median, quartiles, and outliers of a data set.
# Answer:
- Qualitative data: Bar graphs, Pie charts.
- Quantitative data: Histograms (continuous), Bar graphs (discrete), Line graphs, Box - and - whisker plots.

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Explanation:

Question 1: Define "Population" in Statistics

Answer:

Question 2: Define "Sample" in Statistics