Statistics Unit 4: Randomness and Distributions

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Questions and Answers

What do you discover in activity 9-4?

The Empirical Rule

A normal distribution has what kind of shape?

Skewed left
Bell-shaped (correct)
Skewed right
Uniform

What is the formula for calculating the z-score?

z = (x - μ) / σ

Observations above the mean should have ______ z-scores.

positive Signup and view all the answers

Using the empirical rule, what percentage of the data falls within 1 standard deviation of the mean?

68% Signup and view all the answers

What is the definition of standardization?

The process of converting a raw score to a z-score. Signup and view all the answers

What is the definition of the empirical rule?

A rule that states the percentage of data that falls within a certain number of standard deviations from the mean in a normal distribution. Signup and view all the answers

In a normal distribution, what is the name for the 'mean'?

μ Signup and view all the answers

In a normal distribution, what is the name for the 'standard deviation'?

σ Signup and view all the answers

What is the name of the data point that falls within 1 standard deviation above the mean?

μ + σ Signup and view all the answers

What is the name of the data point that falls within 2 standard deviations below the mean?

μ - 2σ Signup and view all the answers

What is the formula used to find the z-score for a specific value in a normal distribution?

z = (x - μ) / σ Signup and view all the answers

Which plot would be used to determine if sample data plausibly came from a normal distribution?

A normal probability plot. Signup and view all the answers

What does the CLT stand for?

Central Limit Theorem Signup and view all the answers

What is the guideline size for a sample size unless a population is extremely non-normal?

n ≥ 30 Signup and view all the answers

What are the three sources of variation?

Bias, Chance Error, and Significant Event. Signup and view all the answers

How does the spread of the sampling distribution of X compare to the spread of the population distribution?

The spread of the sampling distribution of X is smaller than the spread of the population distribution. Signup and view all the answers

What is the symbol for the population mean?

μ Signup and view all the answers

What is the symbol for the sample mean?

X̄ Signup and view all the answers

Is the population mean fixed or variable?

Fixed Signup and view all the answers

Is the sample mean fixed or variable?

Variable Signup and view all the answers

Is the long-term pattern to this variation predictable?

Yes Signup and view all the answers

What other term is given to the "mean of the sample means"?

The expected value of the sample mean Signup and view all the answers

The sampling distribution of the sample means becomes more and more variable as the sample size increases.

False (B) Signup and view all the answers

What conditions have to be met before you can apply the CLT?

The population has to be large (at least ten times larger than the sample size) and the variable of interest has to have a population mean and standard deviation. Signup and view all the answers

What is the difference between statistical significance and statistical confidence?

Statistical significance refers to the probability that the observed result would occur by chance, while statistical confidence refers to the precision of the estimate of a population parameter. Signup and view all the answers

What are the two factors that determine the distance between the population parameter and sample statistic?

The sample size and the confidence level. Signup and view all the answers

The z-score does not indicate how many standard deviations above or below the mean a particular value falls?

False (B) Signup and view all the answers

The sample size increases as the sampling distribution of the sample mean looks more and more like a normal distribution.

True (A) Signup and view all the answers

Describe the properties of normal curves.

Symmetric, mound-shaped, bell-shaped. The mean, median, and mode are equal and in the middle of the distribution. Signup and view all the answers

What is an example of a statistical calculation that can be performed on a normal distribution?

Calculate the probability of a value falling within a specific range or calculating a percentile. Signup and view all the answers

Explain how to calculate the probability of a value falling within a specific range in a normal distribution using a calculator.

Using a calculator's built-in normal CDF (cumulative distribution function) function, you can provide the lower and upper bounds of the range, the mean, and the standard deviation of the distribution. Signup and view all the answers

Explain how to assess whether sample data could come from a normally distributed population based on normal probability plots and graphs.

By examining the normal probability plot, you can look for a straight line pattern, which indicates a normal distribution. Alternatively, you can examine the histogram or dotplot to assess for symmetry, mound shape, and whether the data follows a bell-shaped form suggestive of a normal distribution. Signup and view all the answers

Which of the following would be considered a reasonable estimate for the standard deviation if the range of a set of data is 36 and the data appears mound shaped?

18 (B) Signup and view all the answers

Which of the following scenarios is NOT possible?

All of these are possible. (D) Signup and view all the answers

Match the following graphs of normal distributions with their appropriate mean and standard deviation.

Graph A = Mean 15 and standard deviation 4 Graph B = Mean 13 and standard deviation 2 Graph C = Mean 15 and standard deviation 1 Graph D = Mean 17 and standard deviation 2 Signup and view all the answers

A student received a 540 on their SAT, with a mean of 478 and a standard deviation of 92, what is the minimum score they need to raise their score on their second attempt to improve it?

10 Signup and view all the answers

A symmetrical, mound-shaped distribution has a mean of 42 and a standard deviation of 7. Which of the following is true? (Select one that is correct.)

There are more data values between 42 and 49 than between 28 and 35. (B) Signup and view all the answers

In general, which vary more: averages or individual observations?

Individual observations vary more than averages Signup and view all the answers

Give an intuitive explanation for your answer to the previous question.

If you take a single measurement, it could be influenced by chance or unusual circumstances, leading to a significant deviation from the average. However, when you average multiple measurements, the extreme values tend to cancel each other out, resulting in a more representative and less variable average. Signup and view all the answers

Which vary more: averages based on a few observations or averages based on many observations?

Averages based on a few observations vary more than averages based on many observations Signup and view all the answers

Suppose the IQ scores of students at a certain college follow a normal distribution with a mean of 115 and a standard deviation of 12. Draw a well-labeled sketch of this distribution

A bell-shaped curve centered at 115, with the x-axis representing IQ scores and the y-axis representing the frequency or probability of each score. Label the mean (μ) at 115, and mark the standard deviation (σ) intervals to the left and right of the mean at 103, 127, 91, 139, 79, and 151, respectively. Signup and view all the answers

Shade in the area corresponding to the proportion of students with an IQ less than 100. Based on this shaded region, make an educated guess as to this proportion of students.

Shade the leftmost portion of the bell curve, extending up to the x-value of 100. This area represents the proportion of students with an IQ below 100. A rough estimate based on visually interpreting the shaded area would be around 5% to 10% of the student population. Signup and view all the answers

Use the normal model to determine the proportion of students with an IQ score less than 100.

You would need to calculate a z-score using the formula: (100 - 115) / 12 = -1.25. Then, consult a standard normal distribution table or use a calculator to find the area under the curve to the left of -1.25. This area corresponds to the proportion of students with an IQ less than 100. Signup and view all the answers

Determine the proportion of undergraduates having IQs between 110 and 130.

Calculate the z-scores for both 110 and 130: (110 - 115) / 12 = -0.42 and (130 - 115) / 12 = 1.25. Use a standard normal distribution table or a calculator to find the area under the curve between these two z-scores. This area corresponds to the proportion of students with IQ scores between 110 and 130. Signup and view all the answers

Determine how high a student's IQ must be to be in the top 1% of all IQs at this college.

You would need to find the z-score corresponding to the top 1% of the distribution, which has a probability of 0.99 (99%). Use a standard normal distribution table or a calculator to find this z-score, which will be around 2.33. Then, use the formula: IQ = (z-score * standard deviation) + mean to find the corresponding IQ score. So, IQ = (2.33 * 12) + 115 = 142.96. Students needing an IQ of 143 or higher would be in the top 1% of the distribution. Signup and view all the answers

Find the z* values that cut off the top 5%, top 2.5%, top 1%, and top 0.5% of a standard normal distribution.

You would need to use a standard normal distribution table or a calculator to find the corresponding z-scores. The z* values are roughly: 1.645 for the top 5%, 1.960 for the top 2.5%, 2.326 for the top 1%, and 2.576 for the top 0.5% of the standard normal distribution. These represent the critical z-scores, indicating the cutoff values for the specified proportion of the distribution. Signup and view all the answers

Flashcards

Empirical Rule

For mound-shaped, symmetric distributions, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Z-score

A standardized score indicating how many standard deviations a value is above or below the mean.

What is Standardization?

Converting a value to a standard score, or z-score, so you can compare values from different distributions.

What are the properties of a Normal Distribution?

Normal distributions are bell-shaped, symmetric, and have the mean, median, and mode all equal and in the center of the distribution.