Answer :
To solve the question of checking the conditions for a one-sample z-interval for a proportion [tex]\( p \)[/tex], let's go through each step one by one:
1. Random Condition:
The problem states that we have a random sample of 750 smartphone users. This satisfies the requirement that the data should be collected from a random sample.
2. 10% Condition:
The sample size must be less than 10% of the population of all smartphone users. Although we don't know the exact number of all smartphone users, it is reasonable to assume that there are more than 7,500 smartphone users in total. Therefore, 750 is less than 10% of the total population, meeting this condition.
3. Large Counts Condition:
For a one-sample z-interval, the "Large Counts" condition requires checking both the number of successes and the number of failures. Specifically, these should be greater than or equal to 10.
- Successes: In this context, a "success" is defined as an event that we are interested in. We have 176 reported successes.
- Failures: The number of failures is the sample size minus the number of successes. That means we have [tex]\( 750 - 176 = 574 \)[/tex] failures.
Now, we check if both numbers meet the required threshold:
- Successes: 176 is greater than or equal to 10.
- Failures: 574 is greater than or equal to 10.
Since both numbers (176 successes and 574 failures) are greater than or equal to 10, the Large Counts condition is satisfied.
Therefore, with the conditions satisfied, the appropriate inference procedure to use is a one-sample z-interval for the proportion [tex]\( p \)[/tex].
1. Random Condition:
The problem states that we have a random sample of 750 smartphone users. This satisfies the requirement that the data should be collected from a random sample.
2. 10% Condition:
The sample size must be less than 10% of the population of all smartphone users. Although we don't know the exact number of all smartphone users, it is reasonable to assume that there are more than 7,500 smartphone users in total. Therefore, 750 is less than 10% of the total population, meeting this condition.
3. Large Counts Condition:
For a one-sample z-interval, the "Large Counts" condition requires checking both the number of successes and the number of failures. Specifically, these should be greater than or equal to 10.
- Successes: In this context, a "success" is defined as an event that we are interested in. We have 176 reported successes.
- Failures: The number of failures is the sample size minus the number of successes. That means we have [tex]\( 750 - 176 = 574 \)[/tex] failures.
Now, we check if both numbers meet the required threshold:
- Successes: 176 is greater than or equal to 10.
- Failures: 574 is greater than or equal to 10.
Since both numbers (176 successes and 574 failures) are greater than or equal to 10, the Large Counts condition is satisfied.
Therefore, with the conditions satisfied, the appropriate inference procedure to use is a one-sample z-interval for the proportion [tex]\( p \)[/tex].
