Elijah and Tyler, two high school juniors, conducted a survey on 15 students at their school, asking the students whether they would like the school to offer an afterschool art program, counted the number of “yes” answers, and recorded the sample proportion. 14 out of the 15 students responded “yes”. They repeated this 100 times and built a distribution of sample means. (Note that this question requires having reviewed Section 3.4.2 on the normal approximation to the binomial distribution.) (a) What is this distribution called? ##### Asnwer:
Sampling distribution. (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. ##### Asnwer: (c) Calculate the variability of this distribution and state the appropriate term used to refer to this value. ##### Asnwer: Variability of this distribution is called Standard Error as its basically std dev of the error term(how far the sample mean is from population mean).
n=100
library(psych)
#summary_mean/sqrt(n)The 2010 General Social Survey asked the question: “For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?” Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010. (a) Interpret this interval in context of the data. (b) What does “95% confident” mean? Explain in the context of the application. (c) Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval? (d) If a new survey were to be done with 500 Americans, would the standard error of the estimate be larger, smaller, or about the same. Assume the standard deviation has remained constant since 2010.
Write the null and alternative hypotheses in words and using symbols for each of the following situations. (a) Since 2008, chain restaurants in California have been required to display calorie counts of each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant? (b) Based on the performance of those who took the GRE exam between July 1, 2004 and June 30, 2007, the average Verbal Reasoning score was calculated to be 462. In 2011 the average verbal score was slightly higher. Do these data provide convincing evidence that the average GRE Verbal Reasoning score has changed since 2004?
Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children first counted to 10 successfully. Also provided are some sample statistics.
A food safety inspector is called upon to investigate a restaurant with a few customer reports of poor sanitation practices. The food safety inspector uses a hypothesis testing framework to evaluate whether regulations are not being met. If he decides the restaurant is in gross violation, its license to serve food will be revoked. (a) Write the hypotheses in words. (b) What is a Type 1 Error in this context? (c) What is a Type 2 Error in this context? (d) Which error is more problematic for the restaurant owner? Why? (e) Which error is more problematic for the diners? Why? (f) As a diner, would you prefer that the food safety inspector requires strong evidence or very strong evidence of health concerns before revoking a restaurant’s license? Explain your reasoning.
Determine if the following statements are true or false, and explain your reasoning. If false, state how it could be corrected. (a) If a given value (for example, the null hypothesized value of a parameter) is within a 95% confidence interval, it will also be within a 99% confidence interval. (b) Decreasing the significance level will increase the probability of making a Type 1 Error. (c) Suppose the null hypothesis is μ = 5 and we fail to reject H0. Under this scenario, the true population mean is 5. (d) If the alternative hypothesis is true, then the probability of making a Type 2 Error and the power of a test add up to 1. (e) With large sample sizes, even small di↵erences between the null value and the true value of the parameter, a di↵erence often called the e↵ect size , will be identified as statistically significant.
Four plots are presented below. The plot at the top is a distribution for a population. The mean is 60 and the standard deviation is 18. Also shown below is a distribution of (1) a single random sample of 500 values from this population, (2) a distribution of 500 sample means from random samples of each size 18, and (3) a distribution of 500 sample means from random samples of each size 81. Determine which plot (A, B, or C) is which and explain your reasoning.
It is believed that nearsightedness a↵ects about 8% of all children. In a random sample of 194 children, 21 are nearsighted. (a) Construct hypotheses appropriate for the following question: do these data provide evidence that the 8% value is inaccurate? (b) What proportion of children in this sample are nearsighted? (c) Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution, calculate the test statistic (the Z-statistic). (d) What is the p-value for this hypothesis test? (e) What is the conclusion of the hypothesis test?