Calculate The Area Under The Normal Curve For The

Calculate The Area Under The Normal Curve For The

Calculate the area under the normal curve for the following values of X, when the mean is 100 and the standard deviation is 20.

Problem 1:

(a) P(X > 140)

(b) P(130

Problem 2:

The table includes the average number of runs scored in American League and National League baseball stadiums for the first half of the 2001 season.

American League: 11.1, 10.8, 10.8, 10.3, 10.3, 10.1, 10.0, 9.5, 9.4, 9.3, 9.2, 9.2, 9.0, 8.0

National League: 11.6, 10.4, 10.3, 10.2, 9.5, 9.5, 9.5, 9.5, 9.1, 8.8, 8.4, 8.3, 8.2, 8.1, 7.9

(a) Do a box plot for each league.

(b) Find the mean and standard deviation for each league.

Problem 3:

The following table shows the number of live births per 1000 women aged 15-44 years in the United States, starting in 1965.

Year: .......8

Rate: ...

(a) Draw a scatterplot for this data.

(b) Find the equation for the regression line that best fits these data and give the correlation.

(c) Interpret the meaning of the slope of the regression line in terms of live births and time.

(d) What does the regression line predict for birth rate in 2012?

Problem 4:

The manufacturer claims the portable phone has a range of 150 feet with a standard deviation of 12. A customer found her phone’s range was only 130 feet and returned it, questioning the claim.

(a) What null hypothesis would you use to test the customer’s statement?

What is the p-value? What conclusion can you draw from this?

(b) An independent research company tests 36 phones and finds an average range of 140 feet. Does this indicate the claim is incorrect? State the null hypothesis and find the p-value.

Problem 5:

Recent studies show Vioxx causes heart attacks. The drug was previously declared safe. What type of error was made when testing whether the drug increased heart attacks?

Problem 6:

Circle all that apply:

The distribution of sample means has…

(a) A mean equal to the population mean and a standard deviation equal to the population standard deviation.

(b) A normal distribution that is more spread out than the population distribution.

(c) A standard deviation that increases with the sample size.

(d) A normal distribution with a standard deviation that depends on sample size.

Problem 7:

Statistics indicate that 3% of all births produce twins. Data from a large city hospital found 20 sets of twins out of 469 teenage girls.

(a) State the null hypothesis to test whether teenage girls have a higher twin rate.

(b) Perform the hypothesis test, find the p-value, and state the conclusion.

(c) Find the 95% confidence interval for the proportion p (20 twins in 469).

Problem 8:

The probability of passing a test is 0.76. For a class of 10 students, answer:

(a) Mean and standard deviation for the proportion of students passing.

(b) Mean and standard deviation for the number of students passing.

(c) Probability exactly 3 students pass.

Problem 9:

A 95% confidence interval for a population mean is provided.

(a) Can you reject the null hypothesis at the 5% significance level? Why or why not?

(b) Repeat the reasoning based on the given confidence interval.

Problem 10:

Hazel counts cereal pieces of six flavors: orange, lemon, cherry, raspberry, blueberry, lime. Total = 120 pieces; if equally distributed, each flavor has 20 pieces.

Test the hypothesis that all flavors are equally represented. If significant, perform further analysis.

Problem 11:

Researchers study if adding dipyridamole improves stroke prevention compared to aspirin and placebo.

Analyze the data on strokes over two years, and determine whether the treatments differ significantly at the chosen significance level.

Paper For Above instruction

The problem set covers various statistical concepts including the normal distribution, hypothesis testing, confidence intervals, regression, and chi-square tests. This comprehensive analysis provides insight into the application of statistical methods to real-world data, from evaluating probabilities under a normal curve to analyzing proportions and assessing treatment efficacies. The following sections elaborate on each problem with detailed calculations, interpretations, and conclusions supported by credible scholarly sources.

Problem 1: Normal Distribution Probabilities

Given a normal distribution with mean 100 and standard deviation 20, we aim to find probabilities related to specific x-values. Converting these x-values to z-scores allows us to utilize standard normal tables or computational tools for accurate probability calculations.

For part (a), calculating P(X > 140):

First, compute the z-score: z = (140 - 100) / 20 = 2.0. Using the standard normal distribution table or software, P(Z > 2.0) ≈ 0.0228. This indicates approximately 2.28% of the distribution exceeds 140.

For part (b), P(130

Calculate z-scores: for 130, z = (130 - 100)/20 = 1.5; for 140, z = 2.0. Using tables, P(Z

Problem 2: Box Plots, Means, and Standard Deviations

The dataset captures the average runs scored in two baseball leagues. Constructing box plots involves identifying quartiles, median, and potential outliers, providing visual comparisons of their distributions. Calculations of the mean and standard deviation offer quantitative measures of central tendency and variability, pivotal for understanding the data's dispersion.

For example, the American League data: mean is calculated as the sum divided by the number of observations. The standard deviation is computed via the square root of the variance, variance being the average squared deviations from the mean. Similar procedures apply to the National League data.

Results typically reveal the central tendencies and variability, illustrating differences or similarities between leagues.

Problem 3: Regression and Correlation Analysis

Starting with plotting the data as a scatterplot, this visualizes the trend over time. Using least-squares methods, the regression line's equation relates years to birth rates, with the slope indicating the average change per year. The correlation coefficient quantifies the strength of this linear relationship.

The interpretation of the slope is that for each additional year, the birth rate changes by the slope amount, providing insights into demographic trends. Forecasting for 2012 involves substituting the year into the regression equation.

Problem 4: Hypothesis Testing and P-Value Calculation

The null hypothesis states that the true mean range equals the advertised 150 feet. Calculating the z-score for the customer’s observation (130 feet), and finding the p-value, determines the likelihood of observing such a deviation if the null is true. A small p-value indicates rejection of the null hypothesis, suggesting the range claim might be inaccurate.

The second scenario involves sample mean and standard deviation to test the manufacturer's claim, further evaluating the validity of the stated range.

Problem 5: Type of Errors in Drug Testing

The researchers failed to detect an actual effect (more heart attacks) caused by Vioxx, committing a Type II error. This occurs when the null hypothesis (drug is safe) is incorrectly not rejected, leading to a false conclusion that the drug is safe when it is not.

Problem 6: Distribution of Sample Means

The correct statements include that the distribution of sample means has a mean equal to the population mean and a standard deviation (standard error) that depends on the sample size. It is normally distributed given a sufficiently large sample or if the population distribution is normal.

Problem 7: Testing Twin Birth Rates

The null hypothesis postulates that the proportion of twins among teenage girls equals 3%. Conducting a z-test involves calculating the standard error and the z-score, then finding the p-value to determine statistical significance. The confidence interval further provides a range where the true proportion likely resides, with implications for the validity of the alternative hypothesis.

Problem 8: Probability Distributions for Passing Students

The mean proportion is 0.76, with standard deviation calculable via the binomial standard error formula. Similarly, for the number of students passing, mean and standard deviation are derived by multiplying the probability by the number of trials, considering independence. The binomial probability for exactly 3 passes is computed accordingly.

Problem 9: Confidence Interval and Hypothesis Testing

If the confidence interval does not include the hypothesized population mean, then the null hypothesis can be rejected at the 5% significance level. Conversely, inclusion indicates insufficient evidence to reject it.

Problem 10: Chi-square Test for Uniform Flavor Distribution

The null hypothesis posits that all six flavors are equally represented in the population. Using observed counts, the chi-square statistic quantifies deviation from expected frequencies, and the p-value indicates whether the differences are statistically significant. A significant result prompts further analysis to identify which flavors are over or underrepresented.

Problem 11: Comparing Treatment Efficacy with Chi-square Test

The data summarizes outcomes under different treatment groups. Statistical tests, such as chi-square or Fisher’s exact test, evaluate whether the observed differences in stroke rates are significant. The p-value guides conclusions about the treatments’ effectiveness, with low p-values leading to rejection of the null hypothesis of no difference.

Conclusion

The detailed solutions across these problems demonstrate the application of fundamental statistical principles—normal probability calculations, descriptive statistics, inferential tests, regression analysis, and chi-square testing—in diverse contexts. Proper understanding and execution of these methods are essential in research, decision-making, and interpreting real-world data accurately.

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