a)What is this distribution called? It is called a sampling distribution because the sample size is multiplied to be 100
b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning.
n<- 15
p<- 14/15
n*p## [1] 14n*(1-p)## [1] 1In order for the distribution to be symmetric or normal both of those numbers must be at least 10. Because one of them is less then the distribution is skewed. Taking a look at the data the mean would be greater than the median, meaning that the data would be left skewed.
c) Calculate the variability of this distribution and state the appropriate term used to refer to this value.
sqrt((p*(1-p))/n)## [1] 0.06440612The name of the variability for a sampling distribution is a sampling variability.
d) Suppose that the students were able to recruit a few more friends to help them with sampling, and are now able to collect data from random samples of 25 students. Once again, they record the number of “yes” answers, and record the sample proportion, and repeat this 100 times to build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the original distribution? The variability of the new distribution will be smaller due to the larger sample size, the bigger the sample size is the broader the range of answers is.
a) Interpret this interval in context of the data. According to this data, we have a 95% confidence level that average range of days out of the month where they do not “feel good” is between 3.4 to 4.24.
b) What does “95% confident” mean? Explain in the context of the application. It means that there is a 95% probability that the data has the mean in it, and a 5% chance that it does not.
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? The interval would be smaller
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. The standard error would be larger due to a constant standard deviation being divided by a smaller number.
b)Ho: The GRE Verbal Reasoning score has not changed since 2004 Ha: The GRE Verbal Reasoning score has increased since 2004 Ho:μ=462 Ha:μ>462
a) Are conditions for inference satisfied? Yes because it is a random sample and the data seems almost normal, only slightly skewed
b) Suppose you read online that children first count to 10 successfully when they are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide convincing evidence that the average age at which gifted children first count to 10 successfully is less than the general average of 32 months. Use a significance level of 0.10 Ho:The mean age to count to 10 is 32 months Ha:The mean age to count to 10 is less than 32 months
pnorm(30.69,32,(4.31/sqrt(36)))## [1] 0.0341013P is less than 0.1 so the null hypothesis is rejected
c) Interpret the p-value in context of the hypothesis test and the data. Due to the p-value one can establish that there is evidence tha gifted children can count to 10 at a younger age d) Calculate a 90% confidence interval for the average age at which gifted children first count to 10 successfully.
30.69+qnorm(.90)*4.31/sqrt(36)## [1] 31.6105830.69-qnorm(.90)*4.31/sqrt(36)## [1] 29.76942e) Do your results from the hypothesis test and the confidence interval agree? Explain. Yes because 32 is not in the confidence interval thus the null hypothesis can be refuted this time
a) Write the hypotheses in words. Ho: The restaurant is following regulations Ha: The restaurant is not following regulations
b) What is a Type 1 Error in this context? Type 1 error is when the inspector says the restaurant is not following regulations even though it is
c) What is a Type 2 Error in this context? Type 2 error is when the inspector says the restaurant is following regulations even though it isnt
d) Which error is more problematic for the restaurant owner? Why? Type 1 is more problematic for the owner as their license is revoked when it shouldn’t be e) Which error is more problematic for the diners? Why? Type 2 is more problematic for the diners because they would be dining in a place that does not meet sanitary requirements, believing it does
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. I would prefer the inspector to require very strong evidence before revoking or continuing to allow the restaurant to stay in business. The reason for this is that I do not want mistakes that can end up costing money to the business, and both of those errors would cause the restaurant considerable losses.
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. True, as the 99% interval is an expansion of the 95%
b) Decreasing the significance level will increase the probability of making a Type 1 Error. False, it would decrease the probability of it.
**c) Suppose the null hypothesis is μ = 5 and we fail to reject H0. Under this scenario, the true population mean is 5*.** While it might be it is not entirely true, therefore this statement is false.
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. True,because the power of a test is the probability of not making that type of error.
e) With large sample sizes, even small differences between the null value and the true value of theparameter, a difference often called the effect size , will be identified as statistically significant. True,the larger the sample the more precise the point estimate
a) Construct hypotheses appropriate for the following question: do these data provide evidence that the 8% value is inaccurate? Ho: There is an 8% average of nearsighted children Ha: There isnt an 8% average of nearsighted children b) What proportion of children in this sample are nearsighted?
21/194## [1] 0.1082474Approximately 10.8% 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).
(0.1082474-0.08)/.0195## [1] 1.448585d) What is the p-value for this hypothesis test?
2*(1-pnorm(1.448585))## [1] 0.1474535e) What is the conclusion of the hypothesis test? If the significance level is greater than the p-value the null hypothesis would be rejected.