Textbook Question
In Problems 13–16, construct a confidence interval for p₁ - p₂ at the given level of confidence.
x₁ = 109, n₁ = 475, x₂ = 78, n₂ = 325, 99% confidence
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8. Sampling Distributions & Confidence Intervals: Proportion
Confidence Intervals for Population Proportion
2:40 minutesProblem 8.1.21f
Textbook Question
Reading Rates The reading speed of second-grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm.
f. There is a 5% chance that the mean reading speed of a random sample of 20 second-grade students will exceed what value?
Verified step by step guidance1
Identify the distribution of the sample mean. Since the reading speeds are normally distributed with mean \(\mu = 90\) wpm and standard deviation \(\sigma = 10\) wpm, the sampling distribution of the sample mean for a sample size \(n = 20\) is also normal with mean \(\mu_{\bar{x}} = \mu = 90\) and standard deviation \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{20}}\).
Determine the z-score that corresponds to the upper 5% of the standard normal distribution. This is the z-value \(z_{0.95}\) such that \(P(Z > z_{0.95}) = 0.05\). You can find this value using a standard normal table or a calculator.
Set up the equation relating the sample mean value \(x\) to the z-score: \(z = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}}\). Here, \(x\) is the reading speed value we want to find, which corresponds to the 95th percentile of the sample mean distribution.
Rearrange the equation to solve for \(x\): \(x = z \times \sigma_{\bar{x}} + \mu_{\bar{x}}\).
Substitute the values of \(z\), \(\sigma_{\bar{x}}\), and \(\mu_{\bar{x}}\) into the equation to find the reading speed value that the sample mean will exceed with 5% probability.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve defined by its mean and standard deviation. It models many natural phenomena, including reading speeds, allowing us to calculate probabilities and percentiles for given values.
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Finding Z-Scores for Non-Standard Normal Variables
Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean describes the distribution of means from all possible samples of a given size drawn from a population. It is normally distributed (if the population is normal) with the same mean as the population but a smaller standard deviation called the standard error.
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Standard Error and Critical Values
The standard error measures the variability of the sample mean and is calculated as the population standard deviation divided by the square root of the sample size. Critical values from the normal distribution (z-scores) help determine cutoff points for probabilities, such as the value exceeded with 5% chance.
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