Introductory Statistics

MA207

Day 23 - Introduction to Inference with Averages

Testing Batteries (Adapted from Dunn, P. K. (2013). Comparing the lifetimes of two brands of batteries. JSE, 21(1).)

Energizer AA batteries are designed to provide 1.5 volts. These batteries were tested by loading them into an electronic game that required 250 milliamps of current. Then the experimenters recorded the amounts of time it took before the each battery dropped below a certain voltage level. These times were approximately normally distributed.

• Energizer batteries lasted for an average of 1.35 hours with a standard deviation of 0.044 hours before they dropped below 1.3 volts.
• Energizer batteries lasted for an average of 7.36 hours with a standard deviation of 0.29 hours before they dropped below 1 volt.

Instructor: Put these values on the board for future use.

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Testing Batteries (page 2):

In the study a set of 9 Ultracell batteries (much cheaper) were tested, finding that:

• The Ultracell batteries lasted an average of 1.52 hours before their voltage dropped below 1.3 volts.
• The Ultracell batteries lasted an average of 7.41 hours before their voltage dropped below 1 volt.

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Instructor: Put these values on the board for future use.

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Testing Batteries (page 3)

Does the data on the previous two slides mean that Ultracell batteries are better?�a) No, because we only tested 9 Ultracell batteries and that is not enough data.

b) Yes, because 1.52 is bigger than 1.35 and 7.41 is bigger than 7.36.

c) No, because 1.52 – 1.35 = 0.17, and 7.41 – 7.36 = 0.05 and these are really small numbers.

d) Yes, because 1.52 – 1.35 = 0.17 and that is a lot more than the standard deviation of 0.044.

e) No, because 7.41 – 7.36 = 0.05 and this is a lot less than the standard deviation of 0.029.

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Testing Batteries (page 4)

What null hypothesis should we use?

a) Ultracell batteries are better than Energizer batteries.

b) Energizer batteries are better than Ultracell batteries

c) Ultracell batteries are no different from Energizer batteries.

d) This sample of 9 Ultracell batteries has the same average as the Energizer batteries.

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Simulating Samples of Batteries (1)

Under the null hypothesis we assume that the Ultracell batteries are the same as the Energizer batteries. In the study they took a sample of 9 Ultracell batteries, so let’s simulate that:

• For the 1Volt Energizer data: mean = 7.36 hours, sd = 0.29 hours
• In Excel, build 9 random numbers drawn from this normal distribution:�=NORM.INV( RAND( ) , 7.36 , 0.29 )�(build 9 numbers in this way … probably simplest to do the 9 across a row)
• Find the average of your numbers in a cell to the right of your 9 numbers.
• Pressing “F9” will re-randomize. Press “F9” 10 times and count whether or not you get a mean of at least 7.41 hours.

Simulating Samples of Batteries (2)

What proportion of your random samples of size 9 had an average of at least 7.41 hours?

1. 0
2. 1/10
3. 2/10
4. 3/10
5. 4/10
6. 5/10
7. 6/10

Based on your class’ votes, would you consider “finding an average of 7.41 or more extreme in a sample of 9 batteries” a rare event?

Simulating Samples of Batteries (3)

Now take your random sample of size 9 and fill it down so you actually have 1000 random samples of size 9 along with all of their averages. Once you have this, copy (Control-C) the column of averages and create a histogram in Tinkerplots by:

• In Tinkerplots, drag in a new blank table
• Paste (Control-V) in the data from Excel
• Select the column of data by clicking on its header
• Drag in a new plot and stretch the plot wider
• Drag one of the data points to the side to order them
• Click the “stack” button
• Now, using the divider tool, find the percent of your means that fell above 7.41 hours.

Simulating Samples of Batteries (4)

What percent of these 1000 random sample means were above 7.41 hours?

a) Less than 25%

b) Between 25% and 27.5%

c) Between 27.5% and 30%

d) Between 30% and 32.5%

e) Over 32.5%

Follow up: How do we interpret this percent?

Simulating Samples of Batteries (5)

What did the previous vote tell us?

a) It is very unlikely for groups of 9 Energizer batteries to have a mean above 7.41 hours.

b) It is very unlikely for groups of 9 Ultracell batteries to have a mean above 7.41 hours.

c) It is fairly common for groups of 9 Energizer batteries to have a mean above 7.41 hours.

d) It is fairly common for groups of 9 Ultracell batteries to have a mean above 7.41 hours.

e) We learn nothing.

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Simulating Samples of Batteries (6)

What can we conclude about the time it takes batteries to drop below 1 volt?

a) Ultracell batteries are better than Energizer batteries.

b) Ultracell batteries are no different than Energizer batteries.

c) Ultracell batteries are worse than Energizer batteries.

d) We learn nothing.

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Simulating the 1.3 Volt Data (1)

• Now repeat the simulation to get 1000 random samples of size 9 from a normal distribution with mean 1.35 and standard deviation 0.044.
• Pull the average from each of the 1000 samples into a Tinkerplots histogram.
• Estimate the p-value of getting samples with averages at least 1.52 hours.

What does your simulation tell us about the p-value?

1. The p-value is 0
2. The p-value is less than 0.1
3. The p-value is less than 0.01
4. The p-value is less than 0.001

Simulating the 1.3 Volt Data (2)

What did the previous vote tell us?

a) It is very unlikely for groups of 9 Energizer batteries to have a mean above 1.52 hours.

b) It is very unlikely for groups of 9 Ultracell batteries to have a mean above 1.52 hours.

c) It is fairly common for groups of 9 Energizer batteries to have a mean above 1.52 hours.

d) It is fairly common for groups of 9 Ultracell batteries to have a mean above 1.52 hours.

e) We learn nothing.

Normal Distributions to the Rescue!! (1 Volt Data)

For energizers, the population mean is 7.36 hours with standard deviation 0.29 hours. We got a sample mean of 7.41 hours from 9 Ultracells.

• The standard error of the sampling distribution is: SE = s / sqrt(n) =
• The test statistic (z-score) is:
• In a normal distribution, what is the probability of getting our z-score or larger?�=1 - NORM.DIST( ? ) =

Normal Distributions to the Rescue!! (1.3 Volt Data)

For energizers, the population mean is 1.35 hours with standard deviation 0.044 hours. We got a sample mean of 1.52 hours from 9 Ultracells.

• The standard error of the sampling distribution is: SE = s / sqrt(n) =
• The test statistic (z-score) is:
• In a normal distribution, what is the probability of getting our z-score or larger?�=1 - NORM.DIST( ? ) =