A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. Of the32tires surveyed, the mean lifespan was46,400miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly consistent with the claim?

Note: If you are using a Student'st-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Be careful with the rounding and try to use al decimal places when needed

• Part (a) Multiple Choice
• State the null hypothesis.

a) H₀:?50,000

b) H₀: ?<50,000

c) H₀:?50,000

d) H₀:?= 50,000

• Part (b) Multiple Choice
• State the alternative hypothesis.

a) H_a:?50,000

b) H_a:?50,000

c) H_a:?= 50,000

d) H_a:?<50,000

• Part (c) Multiple Choice
• In words, state what your random variableXrepresents.

a) Xrepresents the average life span, in miles, of the tires surveyed.

b) Xrepresents the number of tires surveyed.

c) Xrepresents the average life span, in years, of the tires surveyed.

d) Xrepresents the life span, in miles, for a tire.

• Part (d)
• State the distribution to use for the test. (Round your answers to two decimal places.)
• X~_______(_______,________)
• Part (e)
• What is the test statistic? (If using thezdistribution round your answers to two decimal places, and if using thetdistribution round your answers to three decimal places.)
• (t or z)=_____________
• Part (f)
• What is thep-value? (Round your answer to four decimal places.)
• P=__________
• Explain what thep-value means for this problem. MultipleChoice

a) IfH₀is true, then there is a chance equal to thep-value that the average life span of a tire is not 46,400 miles or less.

b) IfH₀is false, then there is a chance equal to thep-value that the average life span of a tire is 46,400 miles or less.

c) IfH₀is true, then there is a chance equal to thep-value that the average life span of a tire is 46,400 miles or less.

d) IfH₀is false, then there is a chance equal to thep-value that the average life span of a tire is not 46,400 miles or less.

• Part (g)
• Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to thep-value.
• Part (h)
• Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.
• (i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
• ?=_______
• (ii) Decision:

a) reject the null hypothesis

b) do not reject the null hypothesis

• (iii) Reason for decision:

a) Since?>p-value, we do not reject the null hypothesis.

b) Since?<p-value, we reject the null hypothesis.

c) Since?>p-value, we reject the null hypothesis.

d) Since?<p-value, we do not reject the null hypothesis.

• (iv) Conclusion:

a) There is sufficient evidence to conclude that the average life span of the tires is less than 50,000.

b) There is not sufficient evidence to conclude that the average life span of the tires is less than 50,000.

• Part (i)
• Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to the nearest whole number.)

lower bound;_________ confidience interval/point estimate___________ upper bound____________