Question

1. A city official claims that the proportion of all commuters who are in favor of an expanded public transportation system is 50%. A newspaper conducts a survey to determine whether this proportion is different from 50%. Out of 225 randomly chosen commuters, the survey finds that 90 of them reply yes when asked if they support an expanded public transportation system. Test the official’s claim at α = 0.05.

2. A survey of 225 randomly chosen commuters are asked if they support an expanded public transportation system. 90 said yes. Construct a 95% confidence interval for the proportion of all commuters who support expanded public transportation.

3. An industrial company claims that the mean pH level of the water in a nearby river is 6.8. You randomly select 19 water samples and measure the pH of each. The sample mean and the sample standard deviation are 6.7 and 0.24, respectively. Is there enough evidence at α = 0.05 to reject the claim that the pH level is 6.8? Assume a normally distributed population.

4. You want to estimate the mean pH level of the water in a nearby river. You randomly select 19 water samples and measure the pH of each. The sample mean is 6.7 and the sample standard deviation is 0.24. Construct a 95% confidence interval for the true mean pH level.

5. Explain how using the same sample data and level of significance to conduct a hypothesis test and to construct a confidence interval can lead to the same conclusion. Use your work in #1 − 4 to provide specific examples of the connections between hypothesis tests and confidence intervals.

6. A sample of employees in a certain company gives the following data about salaries. Answer the questions below based on this data set.

$37,000 $95,000 $40,000 $39,750 $37,000 $38,600

(a) Calculate the mean. (b) Find the median.

(c) Find the mode.

(d) Which measure of central tendency you think best represents the data? Explain WHY.

7. Use the following data set to complete the questions below. Calculate the five number summary and sketch a box and whisker plot. Be sure to label your box plot correctly.

78 74 76 80 85 81 62 80 91 46 20 94 95 96 91 82

(a) Calculate the five number summary.

(b) Calculate the interquartile range and determine any potential outliers.

(c) Sketch a boxplot. Be sure to label your boxplot correctly.

(d) Write one inference you can make about this data based on your work above.

8. The lengths of pregnancies are normally distributed with a mean of 267 days and a standard

deviation of 15 days.

(a) Find the probability that an individual woman has a pregnancy shorter than 259 days.

(b) If 36 women are randomly selected, find the probability that they have a mean preg- nancy shorter than 259 days.

(c) There should be a difference in your method for the previous two questions. Explain what you did differently for each problem and explain WHY your answers are different.

(d) Find the probability that an individual woman has a pregnancy longer than 295 days.

(e) Find the probability that an individual woman has a pregnancy between 259 and 295 days.

(f) What is the cutoff number of days for a pregnancy for the top 15% of women?

(g) What is the cutoff number of days for a pregnancy for the bottom 25% of women?

9. In the initial test of the Salk vaccine for polio, 400,000 children were selected and divided into two groups of 200,000. One group was vaccinated with the Salk vaccine while the second group was vaccinated with a placebo. Of those vaccinated with the Salk vaccine, 33 later developed polio. Of those receiving the placebo, 115 later developed polio. Test the claim that the Salk vaccine is effective in lowering the polio rate. Use α = 0.01.

10. Records of randomly selected births were obtained and categorized according to the day of the week that they occurred (based on data from the National Center for Health Statistics). Because babies are unfamiliar with our schedule of weekdays, a reasonable claim is that births occur on the different days with equal frequency. Use a 0.01 significance level to test that claim.

Day Sun Mon Tues Wed Thurs Fri Sat Number of Births 77 110 124 122 120 123 97

11. A study is conducted to find out whether the wait times at two local banks are different. The sample statistics are listed below. Test whether the wait times are the same or different assuming that σ1=σ2. Use α = 0.05.

Bank 1

n1 = 15

x1 = 5.3 minutes s1 = 1.1 minutes

Bank 2

n2 = 16

x2 = 5.6 minutes s2 = 1.0 minutes

  

12. The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lb. Assume that σ is known to be 121.8 lb.

(a) Use a 0.05 significance level to test the claim that the population mean of all such bear weights is greater than 150 lb. Use the usual critical-value method to perform this test.

(b) Find the P -value for this test and explain how the P -value would lead you to the same conclusion that your rejection region method did. (Note that you don’t have to re-do the whole test, but show your work for finding the P -value.)

13. You are given the following regression equation for a scatter plot which The displays data

for x = Weight of Car (in pounds) and y = Miles per Gallon in City: y = −0.006x + 42.825 r2 = 0.7496

(Note: The scatter plot graph is attached to the Canvas assignment as a separate document.)

(a) Find the value of r based on the information given.

(b) Based on your value of r, what conclusion can you make about the correlation of this data?

(c) What does the value of r2 tell you about the regression?

(d) Use the regression equation to estimate miles per gallon for a car that weighs 3000

pounds.

(e) Use the regression equation to estimate the weight of a car that gets 30 miles per gallon.

Answer 1

1. Given : n=225 , X=90

The estimate of the sample proportion is ,

The null and alternative hypothesis is ,

The test is two-tailed test.

The test statistic is ,

The critical values are , ; From Z-table

Decision : Here , the value of the test statistic lies in the rejection region.

Therefore , reject Ho.

Conclusion : There is sufficient evidence to support the claim that the proportion is different from 0.50