Determine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.
Example Problems
A tennis statistician reported that the top-ranked junior player hits aces on of her first serves. Her coach thinks her ace rate is actually higher now, so he took a random sample of of her recent first serves and observed that were aces.
To see how likely a sample like this was to happen by random chance alone, the coach performed a simulation. He simulated 75 samples of first serves from a large population where of first serves were aces. He recorded the proportion of aces in each sample. Here are the counts from his 75 samples:
He wants to test vs. where is her true first-serve ace rate.
Based on these simulated results, what is the approximate -value of the test?
To see how likely a sample like this was to happen by random chance alone, the coach performed a simulation. He simulated 75 samples of first serves from a large population where of first serves were aces. He recorded the proportion of aces in each sample. Here are the counts from his 75 samples:
| Sample proportion | Frequency |
|---|---|
| 0.06 | 1 |
| 0.09 | 3 |
| 0.12 | 9 |
| 0.15 | 17 |
| 0.18 | 18 |
| 0.21 | 15 |
| 0.24 | 8 |
| 0.27 | 3 |
| 0.30 | 1 |
He wants to test vs. where is her true first-serve ace rate.
Based on these simulated results, what is the approximate -value of the test?
A volleyball player's stats sheet from last season showed that of her serves landed in bounds. She suspected her serving had gotten less consistent this season, so she took a random sample of of her recent serves and observed that of them landed in bounds.
To see how likely a sample like this was to happen by random chance alone, the player performed a simulation. She simulated samples of serves from a large population where of serves landed in bounds. She recorded the proportion of in-bounds serves in each sample. Here are the counts from her samples:
She wants to test vs. where is her true serving accuracy.
Based on these simulated results, what is the approximate p-value of the test?
To see how likely a sample like this was to happen by random chance alone, the player performed a simulation. She simulated samples of serves from a large population where of serves landed in bounds. She recorded the proportion of in-bounds serves in each sample. Here are the counts from her samples:
| Sample proportion | Frequency |
|---|---|
| 0.633 | 1 |
| 0.667 | 2 |
| 0.700 | 3 |
| 0.733 | 5 |
| 0.767 | 7 |
| 0.800 | 14 |
| 0.833 | 9 |
| 0.867 | 6 |
| 0.900 | 3 |
She wants to test vs. where is her true serving accuracy.
Based on these simulated results, what is the approximate p-value of the test?
A Minecraft server moderator reported that of players survive their first in-game night. A new player thinks the server's recent update made the game harder, so he took a random sample of recent first-night attempts and observed that survived.
To see how likely a sample like this was to happen by random chance alone, the player performed a simulation. He simulated 100 samples of first-night attempts from a large population where of players survived. He recorded the proportion of survivors in each sample. Here are the counts from his 100 samples:
He wants to test vs. where is the true first-night survival rate.
Based on these simulated results, what is the approximate -value of the test?
To see how likely a sample like this was to happen by random chance alone, the player performed a simulation. He simulated 100 samples of first-night attempts from a large population where of players survived. He recorded the proportion of survivors in each sample. Here are the counts from his 100 samples:
| Sample proportion | Frequency |
|---|---|
| 0.43 | 1 |
| 0.47 | 2 |
| 0.50 | 2 |
| 0.53 | 6 |
| 0.56 | 11 |
| 0.59 | 18 |
| 0.61 | 20 |
| 0.64 | 16 |
| 0.67 | 13 |
| 0.70 | 7 |
| 0.73 | 3 |
| 0.76 | 1 |
He wants to test vs. where is the true first-night survival rate.
Based on these simulated results, what is the approximate -value of the test?
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