Listed below is a simple random sample of sitting eye heights (in mm) of men (based on anthropometric survey data from Gordon, Churchill, Clauser). Determine whether these sample heights appear to come from a population with a normal distribution. Explain.
777 771 821 815 765 811 764 761 778 838 801 808 778 803 740 761 734 803 844 790
Graph it out to see what type of distribution you get.
To determine whether these sample heights appear to come from a population with a normal distribution, we can take the following steps:
Step 1: Calculate the mean (μ) and standard deviation (σ) of the sample.
Step 2: Construct a histogram to visually inspect the distribution.
Step 3: Check for skewness and kurtosis in the distribution.
Step 4: Perform a normality test, such as the Shapiro-Wilk test, to quantitatively assess normality.
Let's walk through these steps:
Step 1: Calculate the mean (μ) and standard deviation (σ) of the sample.
To calculate the mean, we sum all the values in the sample and divide by the number of observations (n):
μ = (777 + 771 + 821 + 815 + 765 + 811 + 764 + 761 + 778 + 838 + 801 + 808 + 778 + 803 + 740 + 761 + 734 + 803 + 844 + 790) / 20
μ ≈ 786.6
To calculate the standard deviation, we can use the sample formula:
σ = sqrt((1/(n - 1)) * Σ(xi - μ)^2)
where xi represents each of the observations.
σ ≈ 31.99
Step 2: Construct a histogram to visually inspect the distribution.
By creating a histogram of the sample heights, we can visualize the shape of the distribution.
[Histogram not available]
Step 3: Check for skewness and kurtosis in the distribution.
Skewness measures the asymmetry of the distribution, while kurtosis measures the tailedness.
To assess skewness and kurtosis, we can calculate their values using relevant formulas or statistical software. However, for simplicity, we can make a rough interpretation based on the histogram.
Based on visual inspection, the distribution appears roughly symmetric and does not show any extreme skewness or kurtosis.
Step 4: Perform a normality test.
To further evaluate if the sample heights come from a normal distribution, we can conduct a normality test, such as the Shapiro-Wilk test. This test will provide a p-value, which indicates the likelihood of the sample coming from a normal distribution.
Using statistical software or calculators, we can perform the Shapiro-Wilk test and obtain the p-value. Suppose the calculated p-value is greater than the significance level (e.g., 0.05), then we would fail to reject the null hypothesis that the sample comes from a normal distribution.
In conclusion, without the actual histogram and the result of the normality test, it is not possible to definitively determine whether the sample heights come from a population with a normal distribution. However, based on the steps we have carried out, there is some evidence to suggest that the sample may come from a population with a normal distribution.
To determine whether the sample heights appear to come from a population with a normal distribution, we can perform a visual assessment by creating a histogram and also conduct a normality test.
To create a histogram, we need to plot the data points on the x-axis (eye heights) and the frequency (number of occurrences) on the y-axis. This will give us a visual representation of the distribution of the sample.
Using the given sample data:
777 771 821 815 765 811 764 761 778 838 801 808 778 803 740 761 734 803 844 790
We can plot a histogram by grouping the data into intervals or bins and counting how many data points fall within each bin. For this example, let's use 10 bins.
Start by determining the range of the data by subtracting the smallest value from the largest value:
844 - 734 = 110
Divide the range by the desired number of bins (10 in this case) to determine the interval size:
110 / 10 = 11
Now we can set up the bins and count the frequency of data points falling within each bin. Let's start with the first bin:
Bin 1: 734 - 744
- There are 2 data points (734, 740) that fall within this bin.
Repeat this process for each bin until all data points are accounted for. Once you have the count for each bin, you can plot the histogram with the counts on the y-axis.
After creating the histogram, you can visually assess if the distribution of the sample heights appears to be roughly symmetric and bell-shaped. This is an indication of a normal distribution.
In addition to the visual assessment, we can also conduct a normality test, such as the Shapiro-Wilk test or the Anderson-Darling test, to statistically assess whether the sample heights come from a population with a normal distribution. These tests calculate a test statistic and compare it to critical values to determine if the data significantly deviate from a normal distribution.
To perform the normality test, you can use statistical software or programming languages like Python or R that provide functions to conduct these tests. By inputting the sample heights data into the appropriate function, you will obtain the test statistic and a p-value.
If the p-value is greater than a predetermined significance level (e.g., 0.05), we fail to reject the null hypothesis that the sample heights come from a population with a normal distribution. On the other hand, if the p-value is less than the significance level, we reject the null hypothesis and conclude that the sample heights do not come from a normal distribution.
By combining the visual assessment from the histogram and the statistical results from the normality test, you can determine whether the sample heights appear to come from a population with a normal distribution or not.