Question

Suppose that \(20 \%\) of the subscribers of a cable television company watch the shopping channel at least once a week. The cable company is trying to decide whether to replace this channel with a new local station. A survey of 100 subscribers will be undertaken. The cable company has decided to keep the shopping channel if the sample proportion is greater than \(.25 .\) What is the approximate probability that the cable company will keep the shopping channel, even though the true proportion who watch it is only \(.20 ?\)

Short answer

The approximate probability that the cable company will keep the shopping channel, even though the true population proportion who watch it is only 20%, is around 0.1056 or 10.56%.

Step-by-step solution

Identify Given Values

The population proportion (p) is given as 0.20 or 20%, the sample size (n) is specified as 100 and the threshold set by the cable company is 0.25 or 25%.

Calculate Sample Mean and Sample Standard Deviation

The sample mean is equal to the population proportion (p), so the sample mean is 0.20 or 20%. The standard deviation \(\(\sigma _{p}\)\) is calculated using the formula \(\sqrt{[p(1-p)/n]}\) where p = 0.20 and n = 100. After calculating, \(\sigma _{p}\)= 0.04

Calculate z-score

The z-score can be calculated using the formula z = \((\bar{p} - p)/ \sigma _{p}\)\), where \(\bar{p}\) is the threshold set by the company (0.25 in this case). Therefore, z is approximately 1.25.

Find probability

The probability that the cable company keeps the channel, even though the true population proportion who watch it is 0.20 can be found by finding the area under the standard normal curve that is greater than the calculated z-value. This probability can be found using standard statistical tables or a calculator which gives the result as approximately 0.1056.

Key concepts

Population Proportion

Understanding the concept of population proportion is crucial when analyzing statistical data, especially for surveys and studies. Imagine a whole school as our population, and we're interested in the number of students who are left-handed. If there are 1,000 students and 200 are left-handed, the population proportion, usually symbolized by a lowercase 'p,' would be 0.20 or 20%.
When companies like the cable one mentioned make decisions based on what they believe to be the preferences of their entire subscriber base, they are using the population proportion to make an educated guess. However, they can't ask every subscriber about their preferences; hence, they rely on a sample to represent the whole population.

Sample Mean

When we talk about the sample mean, we're essentially looking at the average value of a particular trait within a sample. In the context of our problem, the mean is the average proportion of the sample that watches the shopping channel. Since the mean for a proportion is the population proportion, here it is 0.20 or 20%.
We expect our sample to exhibit similar traits to the entire population, which is known as the Law of Large Numbers. As the sample size increases, the sample mean tends to get closer to the population mean. So, when surveying 100 subscribers, we'd estimate the average based on this sample mean.

Standard Deviation

The standard deviation is a measure that tells us how much the values in a set of data deviate from the mean, giving us a sense of the 'spread' of the data. The smaller the standard deviation, the closer the data points are to the mean.
To calculate the standard deviation of the sample proportion, denoted as \( \sigma_{p} \), we use the sqrt{[p(1-p)/n]} formula. This shows us the variability we might expect from the sample proportion due to chance alone, which is vital in determining the reliability of our survey results.

Z-score

A z-score, or standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It tells us how many standard deviations an element is from the mean. In the given problem, the z-score helps us assess how likely it is that the sample proportion differs from the population proportion due to random chance.
Calculated as z = \( (\bar{p} - p) / \sigma_{p} \), where \(\bar{p}\) is the proposed threshold by the cable company (0.25), it lets us understand where this threshold lies in relation to the population proportion.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a perfect normal distribution, the mean, median, and mode are all the same and are at the highest peak of the distribution curve.
When we calculate the z-score, we're assuming that the sample proportions follow a normal distribution, which allows us to find the likelihood of observing a sample proportion above the company's cutoff using the area under the curve. The tail end of the curve (to the right of 1.25 in our z-score case) reflects the approximate probability (0.1056) the cable company will keep the shopping channel, under the assumption that the true population proportion is 0.20.