Question

Please see the questions below from the attached picture.

It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm? Perform a hypothesis test for the population mean at alpha = 0.01.

Please answer the below:

1. Define the null and alternative hypothesis for this test in mathematical terms and in words.(PIC_ONE)
2. Report the level of significance.(PIC_ONE)
3. Include the test statistic and the P-value. See PIC_TWO below. (Note that PIC_TWO shows two tailed P-values. Please report the correct P-value based on the alternative hypothesis.)
4. Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not?

PIC_ONE

# Python methods that calculate confidence intervals require the sample mean and the standard error as inputs. # calculate the sample mean mean = diameters_df['diameters'] .mean() # input the population standard deviation, which was given in Step 1. std_deviation = 0.5000 # calculate standard error = standard deviation / sqrt (n) where n is the sample size. stderr = std_deviation/math.sqrt (len (diameters_df['diameters'])) # construct a 90% confidence interval. conf int 90 = st.norm.interval (0.90, mean, stderr) print("90% confidence interval (unrounded) =", conf int 90) print("90% confidence interval (rounded) = (", round (conf_int_90[0], 2), ",", round (conf_int_90[1], 2), ")") print("") # construct a 99% confidence interval. conf_int_99 = st.norm.interval (0.99, mean, stderr) print("998 confidence interval (unrounded) =", conf_int_99) print("998 confidence interval (rounded) = (", round (conf_int_99[0], 2), ",", round (conf_int_99[1], 2), ")") 90% confidence interval (unrounded) = (2.3984912846323327, 2.6311087153676675) 90% confidence interval (rounded) = ( 2.4, 2.63 ) 99% confidence interval (unrounded) = (2.332661363228155, 2.6969386367718453) 99% confidence interval (rounded) = ( 2.33, 2.7)

PIC_TWO

Answer 1

Question:

Define the null and alternative hypothesis for this test in mathematical terms and in words

Answer:
H0: Null Hypothesis: $\mu$ $\leq$ 2.30 ( The average diameter of ball bearings from this manufacturing process is not greater than 2.30 cm.)

HA: Alternative Hypothesis: $\mu$ > 2.30 ( The average diameter of ball bearings from this manufacturing process is greater than 2.30 cm.) (Claim)

Question:

Report the level of significance

Answer:

The level of significance = $\alpha$ = 0.01

Question:

Include the test statistic and the P-value.

Answer:

The Test Statistic = 3.09

P - Value = 0.002/2 = 0.001

Question:

Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not?

Answer:
Since P - Value = 0.001 is less than $\alpha$ =0.01, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the average diameter of ball bearings from this manufacturing process is greater than 2.30 cm.