Question

Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees0° and standard deviation of 1.00degrees°c. assume 2.32.3​% of the thermometers are rejected because they have readings that are too high and another 2.32.3​% are rejected because they have readings that are too low. draw a sketch and find the two readings that are cutoff values separating the rejected thermometers from the others.

Answer 1

The cutoff values for the rejected thermometers are calculated using the z-scores that correspond to the percentile cutoffs of 2.3% for readings too high and low. With a mean of 0°C and standard deviation of 1.00°C, the cutoff values are approximately 2.00°C and -2.00°C.

Step-by-step explanation:

To find the cutoff values for the thermometer readings, we can assume that the readings are normally distributed with a mean (μ) of 0°C and a standard deviation (σ) of 1.00°C. Since 2.3% of the thermometers are rejected for readings that are too high, and another 2.3% for being too low, we look for the z-scores that correspond to these percentile cutoffs in the standard normal distribution.

First, we need to find the z-score that leaves an area of 2.3% in the upper tail of the normal distribution. We can find this value using a z-table, or using statistical software. Similarly, we find the z-score that leaves an area of 2.3% in the lower tail.

Once we have the z-scores, we use the formula for transforming standard normal distribution values back to the original values which is:

X = (μ + zσ)

For the high cutoff, the z-score is found by looking for 0.977 (100% - 2.3%) in the z-table which roughly corresponds to a z-score of 2. If we round the z-score to 2, to get the temperature reading:

X_high = (μ + zσ) = (0 + 2(1.00)) = 2.00°C

For the low cutoff, we look for the z-score corresponding to 0.023 (2.3%) which is approximately -2. Following a similar process:

X_low = (μ - zσ) = (0 - 2(1.00)) = -2.00°C

Therefore, thermometers with readings above 2.00°C or below -2.00°C would be rejected.

Answer 2

lower cutoff = -2.00°C Upper cutoff = 2.00°C Since the mean is 0 and the standard deviation is 1, your sketch should look EXACTLY like the normal standard deviation curve that you should see in many places online and in text books involving statistics. Now to see where the cutoff points are, look at a standard normal table and look for the value corresponding to your 2.3%. The exact value depends on the table. For instance if it's a "Cumulative from the mean" it will give you the area of the curve from the median point to the desired standard deviation. In which case, you'll see values ranging from [0.0, 0.5). If that's the type of table, convert your 2.3% to 0.5 - 0.023 = 0.477, which if you look up the standard deviation for that value will return you the value 2.00 which means that only 2.3% of the values will lie outside of 2 standard deviations. Since the normal distribution curve is symmetrical, that 2.3% figure will apply on both sides of the mean. E.g. 2.3% of the samples will be above 2 standard deviations above the mean, and 2.5% of the samples will be lower than 2 standard deviations below the mean. Meaning that 4.6% of the overall samples will be further than 2 standard deviations from the mean. In any case, we know that we're rejecting thermometers that are more than 2 standard deviations from the mean. And since our standard deviation is 1.00°C, that means that 2*1.00°C = 2.00°C is how far off the thermometer can read off before getting rejected. So the lower cutoff temperature is -2.00°C and the upper cutoff temperature is 2.00°C. Your sketch should look like a completely normal standard deviation curve and the lower cutoff should be labeled -2.00°C and the upper cutoff 2.00°C where those lines are at the standard deviation values of -2 and 2. If you wish to indicate the set of thermometers being rejected, then shade those regions below -2 deviations and above 2 deviations.