1. This probability distribution shown below relates to the number of times a person goes to a local coffee shop per day. What is the expected mean (average) number of times a person goes to a local coffee shop per day?
2. What is the area equal to under a normal curve distribution?
3. What does the Central Limit Theorem say about the traditional sample size that separates a large sample size from a small sample size?
4. What is inferred by the 95% confidence interval of the mean?
5. What happens to the confidence interval as the sample size increases?
No distribution shown.
1. To calculate the expected mean (average) number of times a person goes to a local coffee shop per day, you need to multiply each number of times by its corresponding probability and sum up the results.
For example, let's say the probability distribution is represented by the following table:
Number of times (x) | Probability (P(x))
------------------------------------
0 | 0.2
1 | 0.3
2 | 0.4
3 | 0.1
To find the expected mean, you need to multiply each number of times by its probability and sum them up:
Expected mean = (0 * 0.2) + (1 * 0.3) + (2 * 0.4) + (3 * 0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4
Therefore, the expected mean number of times a person goes to a local coffee shop per day is 1.4.
2. The area under a normal curve distribution represents the probability of events occurring within a specific range. It can be interpreted as the percentage of the total population or dataset that falls within a certain range of values. The total area under a normal curve is always equal to 1 or 100%.
3. The Central Limit Theorem states that regardless of the shape of the population distribution, as the sample size increases, the distribution of the sample mean approaches a normal distribution. There is no strict cutoff for separating a large sample size from a small sample size based on the Central Limit Theorem. However, it is generally believed that a sample size of 30 or more is considered large enough for the Central Limit Theorem to apply.
4. The 95% confidence interval of the mean provides a range of values within which the population mean is estimated to fall with 95% confidence. This means that if we were to repeat the sampling process multiple times and calculate the confidence interval each time, approximately 95% of those intervals would contain the true population mean.
5. As the sample size increases, the confidence interval becomes narrower. This is because a larger sample size provides more precise estimates of the population mean. With a larger sample size, the sample mean is believed to be a more accurate representation of the population mean, resulting in a smaller margin of error and a narrower confidence interval.