Lori Jeffrey is a successful sales representative for a...

Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on 25$\%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of $.0625 .$ a. How large was the sample used in this analysis? That is, how many sales calls did Lori make during the month? b. Let $\overline{p}$ indicate the sample proportion of book adoptions obtained during the month. Show the sampling distribution of $\overline{p}$ . c. Using the sampling distribution of $\overline{p},$ compute the probability that Lori will obtain book adoptions on 30$\%$ or more of her sales calls during a one-month period.

Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on 25$\%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of $.0625 .$
a. How large was the sample used in this analysis? That is, how many sales calls did Lori make during the month?
b. Let $\overline{p}$ indicate the sample proportion of book adoptions obtained during the month. Show the sampling distribution of $\overline{p}$ .
c. Using the sampling distribution of $\overline{p},$ compute the probability that Lori will obtain book adoptions on 30$\%$ or more of her sales calls during a one-month period.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that as the sample size increases, the distribution of the sample proportions (or means) tends to become normally distributed regardless of the shape of the underlying population distribution. This theorem is the foundation for approximating binomial or other distributions with a normal model in large samples, thereby simplifying probability calculations.

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion represents the distribution of the proportion statistic across many repeated random samples from a population. Under certain conditions (notably when n is large enough), this distribution is approximately normal, with a mean equal to the true proportion and a standard deviation equal to the standard error, making it a central tool in inference.

Normal Approximation in Probability Computation

Once it is established that the sampling distribution of the sample proportion is approximately normal (thanks to the Central Limit Theorem), one can calculate probabilities related to the sample proportion. This is done by standardizing the observed value to a z-score using the formula z = (observed value - mean) / standard error and then using the normal distribution to determine the probability of observing a value as extreme as, or more extreme than, the one in question.

Standard Error for Proportions

The standard error for a proportion quantifies the variability of a sample proportion around the true population proportion. It is computed using the formula SE = sqrt[p(1-p)/n], where p is the true proportion (or assumed proportion under the null hypothesis) and n is the sample size. This metric is crucial for understanding how much sample proportions may fluctuate from one sample to another.

Sample Size Determination

Sample size determination involves rearranging the formula for the standard error to solve for n, given the desired level of precision or a specific observed standard error. By knowing the standard error and the underlying probability p, one can calculate the number of observations required, which is especially valuable in experiments and surveys to ensure reliable estimates.

Transcript

Please give Ace some feedback