A manufacturer of thermostats is concerned that the readings on its thermostats may be showing more variability than expected. The standard deviation of a set of thermostat readings measures the variation between the actual room temperature and the reading displayed on the thermostat. Standard deviation is a common statistic used to determine the typical difference between one particular measurement and the average of a large set of measurements.

In the past, thermostat readings have had a standard deviation of $\sigma = 1.23 ^{\circ} F$. Each day, a random sample of 10 newly manufactured thermostats was selected, and each thermostat was placed in a room that was maintained at 68 degrees F. Compiling the temperature readings for each sample gives a standard deviation identified with the symbol $s$.

The manufacturer wants to see standard deviations of 1.23 degrees F, or less, if possible. Any standard deviation above 1.23 degrees F is a concern, but only standard deviations significantly greater than 1.23 would be reason enough to justify shutting down the production process to perform a quality control check, and make repairs and adjustments, on all the components of the manufacturing plant.

The test statistic:

$$V=(n-1)\cdot (\frac { s }{ \sigma } )^{ 2 }$$

...can be used, where $n$ stands for the number of thermostats tested (10 in this case), to describe the variance between the standard deviation of the sample and the intended standard deviation of the production process (1.23 degrees F).

Historically, the manufacturer has checked the variability of the thermostats once per day. The values of the test statistic $V$ have been compiled for the last 500 days and listed below by frequency:

Interval for V statistic |
Frequency |
---|---|

$0 < V < 2$ | 11 |

$2< V<4$ | 52 |

$4< V<6$ | 63 |

$6< V<8$ | 89 |

$8< V<10$ | 105 |

$10< V<12$ | 81 |

$12< V<14$ | 46 |

$14< V<16$ | 37 |

$16< V<18$ | 12 |

$18< V<20$ | 4 |

TOTAL | 500 |

The manufacturer decides that any value of $V$ that falls in the top 10% of historical data gathered would be high enough to justify a shut down for repairs.

Decide whether the following statements are *true* or *false* about this scenario.

True

False

True

False

On one day in which the sample of 10 thermostats has a standard deviation of s = 1.43 degrees F, the manufacturing process should be shut down for repairs.

True

False

If a sample of 10 thermostats has a standard deviation s that is 30% higher than expected, V would be at least 18.

True

False

It is possible from the historical data to conclude that standard deviations from a sample of 10 thermostats were less than expected approximately half the time.

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