Question

Density of the earth In $1798$, the English scientist Henry Cavendish measured the density of the earth several times by careful work with a torsion balance. The variable recorded was the density of the earth as a multiple of the density of water. Here are Cavendish’s $29$measurements.

a) Enter these data into your calculator and make a histogram. Then calculate one-variable statistics. Describe the shape, center, and spread of the distribution of density measurements.

b) Calculate the percent of observations that fall within one, two, and three standard deviations of the mean. How do these results compare with the $68–95–99.7$rule?

(c) Use your calculator to construct a Normal probability plot. Interpret this plot.

(d) Having inspected the data from several different perspectives, do you think these data are approximately Normal? Write a brief summary of your assessment that combines your findings from (a) through (c).

Short answer

a) Shape: The distribution is essentially symmetric in form.

Center: $5.4479$(mean) or $5.46$(median).

Spread: $0.2209$(Standard deviation).

b) One standard deviations$=-75.86\%$

Two standard deviations$=96.55\%$

Three standard deviations $=100\%$

c)

d) The normal probability map in (C) shows that these measurements are close to normal.

Step-by-step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Explanation

Below are some descriptive statistics and a histogram:

Earth density descriptive statistics

The earth's density readings are generally symmetric, with a mean of $5.45$and a range of $4.88$to $5.85$.

The distribution is essentially symmetric in shape (A portion of the graph is a rough mirror reflection of the other portion.)

The mean is $5.4479$, while the median is $5.46$. (median)

The spread is around $0.2209$. (standard deviation)

Part(b) Step 1: Given Information

Part(b) Step 2: Explanation

The densities closely match the $68-95-99.7$ rule, with $75.86$ percent of the densities ($22$ out of $29$) falling within one standard deviation of the mean. $96.55$ percent of the densities ($28$ out of $29$) are within two standard deviations of the mean, and $100\%$ are within three standard deviations of the mean.

Part(c) Step 1: Given Information

Part(c) Step 2: Explanation

Minitab's normal probability curve is given below:

The Normal probability plot is almost linear, suggesting that the densities are close to Normal.

Part(d) Step 1: Given Information

Part(d) Step 2: Explanation

These measurements are nearly Normal, according to the graphical presentation in (A), the $68-95-99.7$rule check in (B), and the Normal probability plot in (C).