Question

Identify the notation for each given number. For a random sample of 550 randomly selected car batteries, it was found that their output had a mean of 12.8 volts and a standard deviation of 1.5 volts. =550 car batteries =12.8 volts =1.5 volts A population consists of a batch of 26,256 aspirin tablets, and it includes 987 that are defective because they do not meet specifications. A random sample of n=250 of the tablets is obtained and tested, with the result that 19 of them are defective. a. What is the population proportion, p, of defective aspirin tablets?

b. What is the sample proportion, p^ , of defective tablets?
c. How can the discrepancy between the population proportion and the sample proportion be explained?
a. The population proportion of defective aspirin tablets is p= (Round to three decimal places as needed.)

Answer 1

The population proportion is computed as the ratio of defective items to the total population size, while the sample proportion is the ratio of defective items to the sample size. Discrepancies between these proportions arise from sampling variability. For hypothesis testing with unknown population standard deviations, t-distributions or normal distributions are used depending on sample size.

Step-by-step explanation:

The student has presented several statistical situations involving the calculation of proportions, hypothesis testing, and understanding of distributions. Specifically, the student is interested in identifying the correct statistical notations and concepts related to means and proportions of populations and samples, as well as which distributions to use for hypothesis testing.

Example Calculations:

Population proportion (p) is calculated as the number of defective items divided by the total number of items in the population. For the aspirin tablets, p = 987 / 26256.

Sample proportion (p^) is calculated as the number of defective items found in the sample divided by the sample size. Thus for the aspirin sample, p^ = 19 / 250.

The discrepancy between the population proportion and the sample proportion can be due to the variability that is inherent in random sampling.

To assess sample means and conduct hypothesis tests where the population standard deviation is unknown, and the sample sizes are small, the t-distribution should be used. In the case of large samples, even with an unknown population standard deviation, the Central Limit Theorem indicates that the sample means will be approximately normally distributed.

For proportions, when sample sizes are large enough, the normal distribution can be applied for hypothesis testing, due to the properties of the binomial distribution in large samples.