In a statistics activity, students are asked to determine the proportion of times that a spinning penny will land with tails up. The students are instructed to spin the penny 10 times and record the number of times the penny lands tails up. For one student, it lands tails side up six times. The student will construct a 90% confidence interval for the true proportion of tails up. Are the conditions for inference met?

A. Yes, the conditions for inference are met.
B. No, the 10% condition is not met.
C. No, the randomness condition is not met.
D. No, the large counts condition is not met.

The large counts condition for creating a 90% confidence interval is not met in this scenario, as the penny was only spun 10 times, preventing us from expecting at least 10 successes and 10 failures.

Explanation:

To determine if the conditions for inference are met in finding a 90% confidence interval for the proportion of times a spinning penny lands tails up, we must consider the following conditions:

Randomness: The trials (spins) should be random.
Large Counts Condition: This means that we should expect at least 10 successes (tails) and 10 failures (heads). This is based on the np ≥ 10 and n(1-p) ≥ 10 rule, where 'n' is the number of trials and 'p' is the probability of success.
10% Condition: The sample size should be less than 10% of the population when sampling without replacement to ensure the trials are independent.

In this case, the randomness condition is assumed to be met if the student is spinning the penny in a random manner. However, the large counts condition is not met since the penny is only spun 10 times, which does not allow us to expect at least 10 successes and 10 failures. The 10% condition is generally met for practical purposes since the population of possible penny spins is large. Therefore, the correct response is 'no, the large counts condition is not met' because there are not enough trials to satisfy the np ≥ 10 and n(1-p) ≥ 10 rule.