Q.13 - The Practice of Statistics for AP

Question

Cold cabin? The Fathom screen shot below shows the results of taking $500$ SRSs of $10$ temperature readings from a population distribution that’s $N(50, 3)$ and recording the sample variance $2x$ each time.

(a) Describe the approximate sampling distribution. 

(b) Suppose that the variance from an actual sample is $s^{2}x=25$. What would you conclude about the thermostat manufacturer’s claim? Explain 

Step-by-Step Solution

Answer

a). The sampling distribution is right-skewed.

b). The claim appears to be false. 

Part (a) Step 1: Given Information

The results of taking $500$ SRS's of $10$ temperature.

Part (a) Step 2: Explanation

The sampling distribution is right-skewed because the dot plot has the highest columns of dots to the left and a tail to the right.

The sampling distribution is centered at about $9{\left({}^{\circ }F\right)}^2$, because the center of a right-skewed distribution is slightly to the right of the highest points of the graph.

The range of values is from roughly $2{\left({}^{\circ }F\right)}^2$ to $27{\left({}^{\circ }F\right)}^2$.

Part (b) Step 1: Given Information

The results of taking $500$ SRS's of $10$ temperature.

Part (b) Step 2: Explanation

If the manufacturer's claim is true, then the approximate sampling distribution is given by the dot plot.

We note that no dots are above $25$ and only $1$ dot is to its right, this means that it is very unlikely to obtain a sample variance of $25$ and thus it appears that the manufacturer's claim is false.