Cold cabin? The Fathom screenshot 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 minimum each time.

(a) Describe the approximate sampling distribution.

(b) Suppose that the minimum of an actual sample is $40°F$. What would you conclude about the thermostat manufacturer’s claim? Explain.

The Fathom screenshot below shows the results of taking $500$ SRSs of $10$ temperature.

The sample distribution is roughly symmetric. Because the dot plot contains the largest columns of dots roughly in the middle

Because the Centre of a symmetric distribution is roughly at the highest point of the graph, the sampling distribution is centered at around$45°\mathrm{F}$,

The range of values is from roughly $39°\mathrm{F}$to $50°\mathrm{F}$.

The Fathom screenshot below shows the results of taking $500$ SRSs of $10$ temperature

If the manufacturer's claim is correct, the dot plot will show the approximate sampling distribution.

We can see that there are no dots above $40$and only 1 dot to its left, implying that obtaining a sample minimum of $40$is extremely unlikely, and hence the manufacturer's claim appears to be untrue.