Final answer:
The questions relate to using probability distributions and sampling methods in statistics for hypothesis testing and constructing confidence intervals. For large samples, normal approximation is often employed. The correctness of statistical conclusions also depends on the representativeness of the sample, which is determined by the sampling method used.
Step-by-step explanation:
The questions pertain to the subject of statistics, specifically to probability distributions and sampling methods as applied in hypothesis testing and confidence interval estimation. When working with proportions and sample data, statisticians often use the binomial distribution or the normal approximation to the binomial distribution (depending on the sample size and the probability of success) to perform hypothesis tests or to construct confidence intervals.
To perform the hypothesis test for the taste test where 42 percent of respondents are expected to prefer Brand A, and 39 percent actually did in a sample of 100 people, a binomial distribution can be used. However, since the sample size is large, it would be more appropriate to use a normal approximation to the binomial distribution. The steps would include stating the null hypothesis, determining the standard error, calculating the z-score, and comparing it to a critical value or using a p-value approach to make a decision on the hypothesis.
When estimating the percentage of adults in a large city who own cell phones based on a sample of 500 adults where 421 responded positively, a 95 percent confidence interval can be calculated using the formula for a proportion confidence interval, which relies on the standard error and the z-score related to the chosen confidence level. The normal distribution is used because of the large sample size.
The questions also touch upon different sampling methods such as simple random sampling, systematic sampling, convenience sampling, and others. The representativeness of a sample depends largely on how the sample is collected and whether it is likely to accurately reflect the population of interest.