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Chapter 6: Problem 3

Suppose \(5 \%\) of the area under the standard normal curve lies to the left of \(z\). Is \(z\) positive or negative?

Short Answer

Expert verified
The z-score is negative because it is to the left of the mean on the standard normal curve.

Step by step solution

01

Understanding the Standard Normal Curve

The standard normal distribution is a symmetric, bell-shaped curve with a mean of 0 and a standard deviation of 1. In this curve, values to the left of the mean are negative, and values to the right are positive.
02

Determining the Location of the 5% Area

The problem states that 5% of the area under the curve lies to the left of a certain point, which means this area is in the left tail of the distribution. Since 5% is a small percentage and it's on the left side, it indicates a negative z-score.
03

Interpreting the Z-score

In a standard normal distribution, a small percentage (5% in this case) to the left naturally corresponds to negative z-values. This is due to the symmetrical nature of the curve, where equal distributions lie on either side of the mean, which is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-score
A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. When we're dealing with a standard normal distribution, a Z-score tells us how many standard deviations away a particular score is from the mean.
  • If a Z-score is zero, it indicates that the data point's score is identical to the mean.
  • A positive Z-score shows that the data point is above the mean.
  • Conversely, a negative Z-score tells us the value is below the mean.
In plain language, Z-scores help us understand where a value fits in a given statistical distribution. Specifically, they can help us decide if a particular score is exceptional or routine in comparison to the rest of the data. For the problem at hand, since the Z-score results in the small 5% area on the left tail of the normal distribution, the score indeed is negative.
Symmetrical Distribution
The standard normal distribution is a perfect example of symmetrical distribution. This type of distribution is marked by its classic bell shape. Symmetrical distribution means that the data is evenly distributed around the mean. For a standard normal distribution, both halves are mirror images of each other.
The properties of a symmetrical distribution include:
  • Every data point on one side has a counterpart at an equivalent distance from the mean on the opposite side.
  • The mean, median, and mode are all equal and located at the center of the distribution.
  • Because of this symmetry, it can effectively be used to calculate probabilities and establish relationships between data points, like Z-scores.
Understanding the symmetrical nature is crucial in interpreting phenomena like Z-scores, as it explains why certain areas under the curve represent particular percentages, such as the left 5% indicating a negative Z-score.
Left Tail of Distribution
In a standard normal distribution, the tails of the curve taper off indefinitely and contain the extreme values of the data set. The left tail is the part of the distribution that lies to the left of the mean. It represents the lower 50% of the data.
When a percentage, like 5%, lies in this left tail, it signals that a value is significantly below average in this context. If you're dealing with a normally distributed set of data, the left tail includes data points with negative Z-scores.
  • These values occur infrequently compared to the values clustered around the mean.
  • Such a small percentage indicates that the data is relatively uncommon and deviates considerably from the mean.
  • This is the reason we interpret a Z-score that corresponds with the left 5% as a negative value.
Recognizing the role of the left tail is essential for calculating areas under the curve and understanding their implications for statistical analysis.

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Most popular questions from this chapter

Maintenance The amount of money spent weekly on cleaning, maintenance, and repairs at a large restaurant was observed over a long period of time to be approximately normally distributed, with mean \(\mu=\$ 615\) and standard deviation \(\sigma=\$ 42\). (a) If \(\$ 646 dollars is budgeted for next week, what is the probability that the actual costs will exceed the budgeted amount? (b) Inverse Normal Distribution How much should be budgeted for weekly repairs, cleaning, and maintenance so that the probability that the budgeted amount will be exceeded in a given week is only \)0.10 ?$

Find the indicated probability, and shade the corresponding area under the standard normal curve. $$P(-1.78 \leq z \leq-1.23)$$

Inverse Normal Distribution Most exhibition shows open in the morning and close in the late evening. A study of Saturday arrival times showed that the average arrival time was 3 hours and 48 minutes after the doors opened, and the standard deviation was estimated at about 52 minutes. Assume that the arrival times follow a normal distribution. (a) At what time after the doors open will \(90 \%\) of the people who are coming to the Saturday show have arrived? (b) At what time after the doors open will only \(15 \%\) of the people who are coming to the Saturday show have arrived? (c) Do you think the probability distribution of arrival times for Friday might be different from the distribution of arrival times for Saturday? Explain.

Basic Computation: \(\hat{p}\) Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. (a) Suppose \(n=100\) and \(p=0.23 .\) Can we safely approximate the \(\hat{p}\) distribution by a normal distribution? Why? Compute \(\mu_{j}\) and \(\sigma_{\tilde{p}}\) (b) Suppose \(n=20\) and \(p=0.23 .\) Can we safely approximate the \(\hat{p}\) distribution by a normal distribution? Why or why not?

Basic Computation: \(\hat{p}\) Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. (a) Suppose \(n=33\) and \(p=0.21 .\) Can we approximate the \(\hat{p}\) distribution by a normal distribution? Why? What are the values of \(\mu_{\dot{p}}\) and \(\sigma_{\hat{p}} ?\) (b) Suppose \(n=25\) and \(p=0.15 .\) Can we safely approximate the \(\hat{p}\) distribution by a normal distribution? Why or why not? (c) Suppose \(n=48\) and \(p=0.15 .\) Can we approximate the \(\hat{p}\) distribution by a normal distribution? Why? What are the values of \(\mu_{\tilde{p}}\) and \(\sigma_{\hat{p}} ?\)
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