Create (invent/makeup) a data set of n=10 observations with the following properties, sample mean (x with - on top) =4, s=8, g>0 and k>0. You can be accurate up to three decimal points. Without computing it, briefly describe the shape of the histogram.
To create a data set with the given properties, we can follow these steps:
Step 1: Determine the values for g and k (where g>0 and k>0). Let's assume g = 1 and k = 2.
Step 2: Now, we can generate a set of 10 observations using the formula:
Data point = x̄ + s * (g * random number + k).
Step 3: Generate 10 random numbers between -1 and 1 (inclusive). Let's say the random numbers are: -0.123, 0.456, -0.789, 0.321, -0.654, 0.987, 0.135, -0.864, 0.246, -0.579.
Step 4: Plug in these random numbers into the formula:
Data point = 4 + 8 * (1 * random number + 2).
Calculating the values for each data point, we get:
Data point 1 = 4 + 8 * (1 * (-0.123) + 2) ≈ 21.016
Data point 2 = 4 + 8 * (1 * 0.456 + 2) ≈ 19.648
Data point 3 = 4 + 8 * (1 * (-0.789) + 2) ≈ 8.072
Data point 4 = 4 + 8 * (1 * 0.321 + 2) ≈ 11.768
Data point 5 = 4 + 8 * (1 * (-0.654) + 2) ≈ 4.768
Data point 6 = 4 + 8 * (1 * 0.987 + 2) ≈ 21.896
Data point 7 = 4 + 8 * (1 * 0.135 + 2) ≈ 8.080
Data point 8 = 4 + 8 * (1 * (-0.864) + 2) ≈ -11.096
Data point 9 = 4 + 8 * (1 * 0.246 + 2) ≈ 8.968
Data point 10 = 4 + 8 * (1 * (-0.579) + 2) ≈ 2.968
Therefore, the data set with the given properties is:
{21.016, 19.648, 8.072, 11.768, 4.768, 21.896, 8.080, -11.096, 8.968, 2.968}
Now, let's consider the shape of the histogram without computing it:
Based on the properties of the data set, we know that the mean is 4, which suggests that the data is centered around 4. Additionally, the sample standard deviation is 8, indicating a relatively large spread of data points.
Since we are not computing the exact data values, it is difficult to determine the exact shape of the histogram. However, given the properties mentioned, we can expect the histogram to have a somewhat symmetric or bell-shaped distribution. This is because the mean of 4 suggests that data points are likely to be concentrated around this value, and the standard deviation of 8 implies some variability in the distribution.
To get a more accurate understanding of the histogram's shape, it would be necessary to compute the actual data values and construct the histogram using a larger sample size.