Question

The proportion of individuals with an Rh-positive blood type is \(85 \%\). You have a random sample of \(n=500\) individuals. a. What are the mean and standard deviation of \(\hat{p}\), the sample proportion with \(\mathrm{Rh}\) -positive blood type? b. Is the distribution of \(\hat{p}\) approximately normal? Justify your answer. c. What is the probability that the sample proportion \(\hat{p}\) exceeds \(82 \% ?\) d. What is the probability that the sample proportion lies between \(83 \%\) and \(88 \%\) ? e. \(99 \%\) of the time, the sample proportion would lie between what two limits?

Short answer

Answer: The probability that the sample proportion lies between 83% and 88% is approximately 0.8352 or 83.52%.

Step-by-step solution

Calculate the mean and standard deviation of the sample proportion

Given the population proportion (p) of Rh-positive blood type is 85%, or \(p=0.85\). The sample size (n) is 500. The mean of the sample proportion (\(\hat{p}\)) is equal to the population proportion: \(\mu_{\hat{p}} = p = 0.85\) The standard deviation of the sample proportion is given by the formula: \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\) Plugging in the values, we get: \(\sigma_{\hat{p}}=\sqrt{\frac{0.85\times(1-0.85)}{500}}=\sqrt{\frac{0.85\times0.15}{500}}\approx0.0173\)

Check if the distribution is approximately normal

To determine whether the distribution is approximately normal or not, we will use the following conditions for a binomial distribution: 1. np ≥ 10 2. n(1-p) ≥ 10 Calculating these values, we find: 1. np = 500 × 0.85 = 425 2. n(1-p) = 500 × 0.15 = 75 Since both values are greater than 10, the distribution of \(\hat{p}\) is approximately normal.

Calculate the probability that the sample proportion exceeds 82%

We can calculate the probability using the z-score formula and the standard normal table. First, find the z-score for \(\hat{p}=0.82\): \(z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.82-0.85}{0.0173}\approx-1.73\) The probability of the sample proportion exceeding 82% is equal to the probability of z-score being greater than -1.73, which, according to the standard normal table, is about 0.9582.

Calculate the probability that the sample proportion lies between 83% and 88%

First, find the z-scores for \(\hat{p}=0.83\) and \(\hat{p}=0.88\): \(z_1=\frac{0.83-0.85}{0.0173}\approx-1.16\) \(z_2=\frac{0.88-0.85}{0.0173}\approx1.73\) Now, we can calculate the probability by finding the area under the standard normal curve between -1.16 and 1.73. Using the standard normal table, we find the probabilities for each z-score: \(P(z_1)\approx0.1230\) \(P(z_2)\approx0.9582\) The probability that the sample proportion lies between 83% and 88% is equal to the difference between these probabilities: \(P(0.83\leq\hat{p}\leq0.88) = P(z_1) - P(z_2) = 0.9582 - 0.1230 \approx 0.8352\)

Find the limits for 99% of the sample proportion

To find the limits for 99% of the sample proportion, we can use the z-scores corresponding to the 0.005 (lower limit) and 0.995 (upper limit) percentiles of the standard normal distribution. These z-scores are approximately -2.576 and 2.576. Next, we can use the z-score formula to find the limits: Lower limit: \(\hat{p}_L=\mu_{\hat{p}}+z_L\times\sigma_{\hat{p}}=0.85+(-2.576)\times0.0173\approx0.8055\) Upper limit: \(\hat{p}_U=\mu_{\hat{p}}+z_U\times\sigma_{\hat{p}}=0.85+(2.576)\times0.0173\approx0.8945\) Therefore, 99% of the time, the sample proportion lies between 80.55% and 89.45%.

Key concepts

Normal Distribution

Understanding normal distribution is crucial for studying statistics as it describes how data is distributed around the mean of a dataset. It is bell-shaped and symmetric about the mean. When we're analyzing the distribution of sample proportions, this concept helps us determine if the data follows a predictable pattern. In practical terms, for a dataset to be considered "normally distributed," about 68% of the data should fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three.

When we're working with sample proportions, like the proportion of individuals with Rh-positive blood, ensuring that data follows a normal distribution means the results from our sample should closely estimate the real characteristics of the entire population. If certain conditions are met, specifically large enough sample sizes, the sample proportion's distribution can be approximated by a normal distribution. This approximation allows us to make predictions and inferences about the population from which the sample is drawn.

Sample Proportion

The sample proportion is a fundamental concept when dealing with probability and statistics. In simple terms, it refers to the fraction or percentage of individuals in a sample with a specific characteristic, like having Rh-positive blood in our case.

Denoted by \hat{p}, the sample proportion gives us an estimate of the true population proportion (denoted by \(p\)). When working with proportions, it’s important to have a large enough sample size to ensure accuracy. The larger the sample, the closer the sample proportion is expected to be to the population proportion due to the Law of Large Numbers.

• Helps to estimate the true proportion in the whole population.
• Relies on sample size; a bigger size means more reliable estimates.
• Often accompanied by calculations to find its mean and standard deviation, enabling further analysis like making predictions with certain confidence levels.

Z-score

The z-score is a tool that helps us understand how far a point is from the mean of a dataset, measured in standard deviations. In the context of sample proportions, a z-score allows us to determine the probability of a sample proportion occurring, providing insights on likelihood and variation.

Mathematically, it is calculated with the formula:\[ z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} \]where \hat{p} is the observed sample proportion, \(\mu_{\hat{p}}\) is the mean of the sample proportion, and \(\sigma_{\hat{p}}\) is the standard deviation of the sample proportion.

• Helps to convert the sample proportion into a standard normal form.
• Enables us to use standard normal distribution tables to find probabilities.
• Essential for assessing whether observed results deviate noticeably from expected results.

Calculating the z-score is a key step in hypothesis testing, allowing us to assess the significance of our results.

Standard Deviation

Standard deviation is a measure that describes the amount of variation or dispersion in a set of data values. In statistics, it's a critical tool for understanding how "spread out" the data is relative to the mean.

When dealing with sample proportions, the standard deviation helps us assess the extent to which the sample proportion might vary if we repeated the same sampling process multiple times.

• Calculated for a sample proportion using the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]where \(p\) is the population proportion and \(n\) is the sample size.
• Gives insight into the reliability and stability of statistics derived from the sample.
• A smaller standard deviation means the data points are closer to the mean, signifying less variation or spread.

Understanding standard deviation is vital for interpreting data distributions and for making predictions based on statistical figures, ensuring accurate decision-making.