Use the relatively small number of given bootstrap samples to construct theconfidence interval. In a Consumer Reports Research Center survey, women were asked if theypurchase books online, and responses included these: no, yes, no, no. Letting"yes" = 1 and letting "no" = 0, here are ten bootstrap samples for thoseresponses: \(\\{0,0,0,0\\},\\{1,0,1,0\\}\\{1,0,1,0\\},\\{0,0,0,0\\},\\{0,0,0,0\\},\\{0,1,0,0\\},\\{0,0,0,0\\},\\{0,0,0,0\\},\\{0,1,0,0\\},\\{1,1,0,0\\}.\) Using only the ten given bootstrap samples, construct a \(90 \%\) confidenceinterval estimate of the proportion of women who said that they purchase booksonline.
Short Answer
Expert verified
The 90% confidence interval is 0 to 0.5.
Step by step solution
01
Convert Responses to Numerical Values
Replace 'yes' with 1 and 'no' with 0 for the given responses.
To determine the 90% confidence interval from the sorted proportions, remove the lowest 5% and the highest 5%. Since there are 10 samples, 10% of 10 is 1. Thus, remove the lowest and highest one value. The confidence interval is the range of the remaining proportions: \( \{0, 0, 0, 0, 0, 0.25, 0.25, 0.5\} \).
06
Identify the Interval
The 90% confidence interval estimates for the proportion of women who purchase books online is from 0 to 0.5.
These are the key concepts you need to understand to accurately answer the question.
sampling distribution
Imagine you asked a group of women whether they purchase books online. Their responses might vary. If you surveyed a new group each time, you’d also get different outcomes. This variation is what we call the sampling distribution. It showcases the range of possible results we might see if we took multiple random samples from the same population.
In our exercise, instead of surveying many groups, we used bootstrap samples (re-sampling with replacement of the original group). The proportions from these samples create an approximate sampling distribution. For example, one bootstrap sample gave us a proportion of 0%, another 25%, and another 50%. This range represents different results we might get if we surveyed different groups of women multiple times. This method helps us understand how variable our estimates can be and explains why no single survey gives a complete picture.
proportion estimate
When we talk about a proportion estimate, we're referring to the fraction or percentage of a whole. Here, we're estimating the proportion of women who purchase books online.
From the original responses, we get: 'no', 'yes', 'no', 'no'—numerically, that's 0, 1, 0, 0. If we calculate the proportion of 'yes' responses, which represent book purchases, we'd get \(\frac{1}{4} = 0.25\) or 25%.
However, in our exercise, we go beyond the initial group and use bootstrap samples to estimate. With bootstrap sampling, we re-sample the original responses multiple times to mimic surveying more groups. We then calculate the proportion of 'yes' (purchase) in each sample. For instance, one bootstrap sample might be \[1, 0, 1, 0\] or 50%, and another might be \[0, 0, 0, 0\] or 0%. These varied proportions help us better understand the possible range of our proportion estimate.
confidence interval
A confidence interval gives us a range in which we expect the true proportion to lie, with a certain level of confidence (like 90%, 95%, etc.). It reflects how sure we are about where the 'true' proportion falls, based on our sample data.
Let's say we want a 90% confidence interval for our survey question. We first gather several bootstrap samples and calculate the proportion of 'yes' for each. We then order these proportions. To form a 90% confidence interval, we remove the lowest 5% and the highest 5% of our ordered proportions. For our exercise, we had 10 samples, and removing the top and bottom values leaves us with the 90% middle range. This range is our 90% confidence interval.
In our specific case, after sorting the proportions, we removed the extreme values. Our remaining range, from 0 (0%) to 0.5 (50%), is our 90% confidence interval. This suggests that we're 90% confident the true proportion of women who purchase books online lies somewhere between 0 and 50%.
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Which of the following best explains why the collapsing cloud should form a disk? Colliding cloud particles exchange angular momentum and, on average, end up with the rotation pattern for the cloud as a whole.
the law of conservation of angular momentum The nebular theory also predicts that the cloud should heat up as it collapses. What physical law explains why it heats up? the law of conservation of energy The nebular theory also predicts that the cloud will flatten into a disk as it shrinks in size.
The conservation of angular momentum is crucial in the formation of a star. Random collisions between particles in the contracting cloud cause it to flatten into a disk. (A) All stars form from clouds of gas and dust.
The discovery that terrestrial planets and Jovian planets orbit in opposite directions around a star would most challenge the nebular theory, as it contradicts the expectation of a cohesive direction for planetary orbits.
A cloud may start with any size or shape, and different clumps of gas within the cloud may be moving in random directions at random speeds. When the cloud collapses, these different clumps collide and merge, resulting in a flattened rotating disk.
Collapsing clouds refer to the process where a cloud of gas and dust undergoes gravitational collapse, leading to the formation of dense cores that eventually give rise to stars or stellar systems.
Increasing temperatures in the shrinking nebula vaporized most of the solid material that was originally present. At the same time, the collapsing nebula began to rotate faster through the conservation of angular momentum (see the Orbits and Gravity).
Starting point: A cloud of interstellar gas and dust, the "solar nebula"; Most of it (98%) is hydrogen and helium, but it includes atoms and dust grains of heavier material, formed in previous generations of stars.
“nebula” means cloud. The solar system formed from a large cloud of gas that collapsed under the force of its own gravity. Its rotation increasing as it collapsed, via the conservation of angular momentum.
Summary: The terrestrial planets formed close to the Sun where temperatures were well suited for rock and metal to condense. The jovian planets formed outside what is called the frost line, where temperatures were low enough for ice condensation.
Explanation: All of the planets rotate in the same direction. This can be explained by the nebular disk spinning in all the same direction. All of the planets rotate is essential the same plane of rotation.
The main problem involved angular momentum distribution between the Sun and planets. The planets have 99% of the angular momentum, and this fact could not be explained by the nebular model. As a result, astronomers largely abandoned this theory of planet formation at the beginning of the 20th century.
Another problem with the nebular hypothesis was the fact that, whereas the Sun contains 99.9 percent of the mass of the solar system, the planets (principally the four giant outer planets) carry more than 99 percent of the system's angular momentum.
The nebular hypothesis assumes all nine planets were created 4.5 billion years ago (Ga) as molten bodies that cooled with the same size and chemical composition they have today. Reevaluation of the nebular hypothesis proves it has been wrong since its inception.
The solar nebular hypothesis describes the formation of our solar system from a nebula cloud made from a collection of dust and gas. It is believed that the sun, planets, moons, and asteroids were formed around the same time around 4.5 billion years ago from a nebula.
Heavier elements sank to the center, forming iron-rich cores. Lighter materials were buoyed upward to form the outer rocky layers. The slowly rotating solar nebula collapsed under its own gravity to form a rapidly rotating disk, with the Sun at the center.
Explanation: The law that explains why a collapsing cloud usually forms a protostellar disk around a protostar is the Conservation of Angular Momentum.
The force of gravity tends to make things into spheres, which is why planets and stars are round. However, spinning flattens objects out: gas clouds, stars in spiral galaxies, and other systems form disks under rapid rotation. Stars form out of clouds of gas, which slowly collapse under the force of gravity.
As the cloud collapsed, it began to spin faster and faster due to the conservation of angular momentum, which caused it to flatten into a disk. The disk eventually became the protoplanetary disk from which the planets and other bodies in the solar system formed.
Introduction: My name is Edwin Metz, I am a fair, energetic, helpful, brave, outstanding, nice, helpful person who loves writing and wants to share my knowledge and understanding with you.
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